The angular acceleration is -1.43 rev/s². The number of revolutions made by the fan blades is -1870.5 rev. It will take an additional 125.87 seconds for the fan to come to rest.
Explanation:To find the angular acceleration, we can use the formula:
Angular acceleration = (final angular velocity - initial angular velocity) / time
Substituting the given values:
Angular acceleration = (180 rev/min - 550 rev/min) / 4.30 s = -1.43 rev/s²
For part B, the number of revolutions made by the fan blades can be found using the formula:
Number of revolutions = (final angular velocity - initial angular velocity) * time
Substituting the given values:
Number of revolutions = (180 rev/min - 550 rev/min) * 4.30 s = -1870.5 rev
For part C, the time required for the fan to come to rest can be found using the formula:
Time = (final angular velocity - initial angular velocity) / angular acceleration
Substituting the given values:
Time = (0 rev/min - 180 rev/min) / -1.43 rev/s² = 125.87 s
A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed v1 = + 0.70 c in one direction, and the other travels at a speed v2 = − 0.76 c in the other direction, as observed from the spaceship. Part A What speed does the first escape pod measure for the second escape pod? Express your answer using two significant figures.
To develop this problem it is necessary to apply the related concepts at relative speed.
When an observer perceives the relative speed of a second observer, the function is described,
[tex]v' = \frac{v_1-v_2}{1-\frac{v_1v_2}{c^2}}[/tex]
Where,
[tex]v_1[/tex] = The velocity of the first escape pod
[tex]v_2[/tex] = The velocity of the second escape pod
c = Speed of light
v' = Speed of the first escape pod relative to the second escape pod.
Our values are given as,
[tex]v_1[/tex]= 0.7c
[tex]v_2[/tex]= -0.76c
Replacing we have,
[tex]v' = \frac{v_1-v_2}{1-\frac{v_1v_2}{c^2}}[/tex]
[tex]v' = \frac{0.7c-(-0.76c)}{1-\frac{(0.7c)(-0.76c)}{(3*10^8)^2}}[/tex]
[tex]v' = \frac{0.7c-(-0.76c)}{1-\frac{(0.7c)(-0.76c)}{(3*10^8)^2}}[/tex]
[tex]v' = 2.85*10^8m/s[/tex]
Therefore the speed of the first escape pod measure for the second escape pod is [tex]v' = 2.85*10^8m/s[/tex]
The relative speed of the second escape pod as measured by the first escape pod, using relativistic velocity addition, is approximately -0.04c (or just 0.04c considering the magnitude), where c is the speed of light.
Explanation:The question involves calculating the relative speed of one escape pod as observed by the other escape pod in a scenario where they are moving in opposite directions. To do this, we must use the formula for relativistic velocity addition. The formula is as follows:
V = (v1 + v2) / (1 + v1*v2/c²),
where V is the relative velocity as measured by one escape pod, v1 and v2 are the velocities of the escape pods, and c is the speed of light. For this particular problem:
v1 = +0.70c (the velocity of the first escape pod as observed from the spaceship)v2 = -0.76c (the velocity of the second escape pod as observed from the spaceship, note the negative sign because it is in the opposite direction)We substitute the values into the relativistic velocity addition formula to find the relative speed:
V = (0.70c - 0.76c) / (1 - 0.70*(-0.76)c²/c²)
Doing the calculations:
V = -0.06c / (1 + 0.532)c²/c²)
V = -0.06c / 1.532
V = -0.039c ≈ -0.04c
The negative sign indicates that the second escape pod is moving in the direction opposite to the first pod as measured by the first pod. It's important to keep in mind that this result is only an approximation, rounded to two significant figures as per the question's request.
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In this example we will use pendulum motion to actually measure the acceleration of gravity on a different planet. An astronaut on the surface of Mars measures the frequency of oscillation of a simple pendulum consisting of a ball on the end of a string. He finds that the pendulum oscillates with a period of 1.5 s. But the acceleration due to gravity on Mars is less than that on earth, gMars=0.38gearth. Later, during a journey to another planet, the astronaut finds that his simple pendulum oscillates with a period of 0.92 s. What planet is he now on?SOLUTIONSET UP Each planet has a different value of the gravitational acceleration g near its surface. The astronaut can measure g at his location, and from this he can determine what planet he's on. First we use the information about Mars to find the length L of the string that the astronaut is swinging. Then we use that length to find the acceleration due to gravity on the unknown planet.
Answer:
Explanation:
Let length of the pendulum be l . The expression for time period of pendulum is as follows
T = 2π[tex]\sqrt{\frac{l}{g} }[/tex]
For Mars planet ,
1.5 = [tex]2\pi\sqrt{\frac{l}{.38\times9.8} }[/tex]
For other planet
.92 = [tex]2\pi\sqrt{\frac{l}{g_1} }[/tex]
Squiring and dividing the two equations
[tex]\frac{1.5^2}{.92^2} = \frac{g_1}{3.8\times9.8}[/tex]
[tex]g_1 = 9.9[/tex]
The second planet appears to be earth.
A 1-kg ball is released from a height of 6 m, and a 2-kg ball is released from a height of 3 m. Air resistance is negligible as they fall. Which of the following statements about these balls are correct?
A) Both balls will reach the ground with the same kinetic energy.
B) Both balls will reach the ground with the same speed.
C) As they reach the ground, the 1-kg ball will be moving faster than the 2-kg ball.
D) As they reach the ground, the 1-kg ball will have more kinetic energy than the 2-kg ball because it was dropped from a greater height.
E) Both balls will take the same time to reach the ground.
Answer:
A) True B) false C) True D) False E) False
Explanation:
A) We know that the potential energy is transformed into kinetic energy since the ball is released.
[tex]EP1=m*g*h = 1 * 9.81 * 6 = 58.86J\\[/tex]
This energy will be the same at the moment that the ball is about to hit the floor. Therefore there was a transformation from potential to kinetic energy
EK1 = EP1
[tex]EK=\frac{1}{2}*m*v^{2} \\v1=\sqrt{(58.86*2)/1} \\v1=10,84 m/s[/tex]
With the 2kg ball happens the same
[tex]EP2=m*g*h2=2*9.81*3=58.86J\\[/tex]
Velocity at the moment when is about to hit the floor (due to the energy transform) will be:
[tex]EK2=\frac{1}{2}*m*v^{2} \\v1=\sqrt{(58.86*2)/2} \\v2=7.67 m/s[/tex]
A) Since both balls have the same potential energy, they will have the same kinetic energy
B) The velocities are differents as they were calculated before.
C) Therefore the 1 kg ball moves faster as it reaches the ground.
D) The kinetic energy is the same for both balls as it was calculated before.
E) The 2 kg ball will reach the ground first, since it is closer to the ground
With the velocities calculated in the previous steps we have now:
[tex]1 kg ball\\v=v0+g*t\\t1=10.85/9.81 = 1.1s\\\\2 kg ball\\t2=7.672/9.81 = 0.782s[/tex]
Your friend is catching a falling basketball after it has passed through the basket. Her hands move straight down while catching the ball. It takes about 0.10 s for the player to lower her hands to stop the ball. Assume the mass of the ball is 0.60 kg, and that the ball has fallen a vertical distance of 1.2 m before reaching the player's hand. Determine the average force that her hands exert on the ball while catching it.
To determine the average force exerted by her hands on the ball while catching it, we can use the equation F = m * a. First, we find the initial velocity at the top of the ball's trajectory. Then, we calculate the acceleration using the change in velocity and time. Finally, we can find the force using the mass and acceleration.
Explanation:To determine the average force that her hands exert on the ball while catching it, we can use the equation F = m * a, where F is the force, m is the mass of the ball, and a is the acceleration. First, we need to find the acceleration of the ball. We can use the equation a = Δv / Δt, where Δv is the change in velocity and Δt is the change in time. Since the ball stops in 0.10 s, the change in velocity is equal to the initial velocity, which is the velocity at the top of the ball's trajectory. Using the equation v = u + a * t, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken, we can find the initial velocity at the top of the trajectory. The initial velocity can be determined using the equation v² = u² + 2as, where s is the vertical distance traveled. Substituting the known values, we can find the initial velocity. Once we have the initial velocity, we can use it to calculate the acceleration, and finally the force.
Let's go step by step. First, we need to find the initial velocity at the top of the trajectory. Using the equation v² = u² + 2as, where v is the final velocity (0 m/s at the top), u is the initial velocity, a is the acceleration (-9.8 m/s²), and s is the vertical distance traveled (1.2 m), we can solve for u: 0 = u² + 2(-9.8 m/s²)(1.2 m). Solving for u, we get u = 3.13 m/s. Now that we have the value of the initial velocity, we can find the acceleration using the equation a = (v - u) / t, where v is the final velocity (0 m/s), u is the initial velocity (3.13 m/s), and t is the time taken (0.10 s). Substituting the values, we get a = (0 m/s - 3.13 m/s) / 0.10 s = -31.3 m/s². Finally, we can calculate the force using the equation F = m * a, where m is the mass of the ball (0.60 kg) and a is the acceleration (-31.3 m/s²). Substituting the values, we get F = (0.60 kg)(-31.3 m/s²) = -18.78 N. Since force is a vector quantity, it is important to note that we have considered the negative sign in the calculation, which indicates that the force is directed upwards.
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The force exerted is approximately 29.1 N in the opposite direction of the ball's motion.
Determining the Average Force Exerted on a Falling Basketball
To find the average force exerted by the player's hands while catching the falling basketball, we need to understand a few key physics concepts involving motion and force.
Steps to Calculate the Average Force:
Calculate the velocity of the ball just before it is caught: Using the equation for free fall, we can determine the final velocity (v) of the ball just before it reaches the player's hands:
[tex]v^2 = u^2[/tex]+ 2gh, where u = initial velocity (0 m/s), g = acceleration due to gravity (9.8 m/s²), and h = height (1.2 m)
[tex]v^2[/tex] = 0 + 2(9.8)(1.2)
[tex]v^2[/tex]= 23.52
v = [tex]\sqrt{23.52}[/tex]≈ 4.85 m/s
Calculate the change in momentum: The change in momentum, or impulse, occurs as the ball is caught and brought to a stop in the player's hands. The ball's mass is 0.60 kg.
Impulse = change in momentum = m x (v - u)
Impulse = 0.60 kg x (0 m/s - 4.85 m/s)
Impulse = -2.91 kg·m/s
Determine the average force: We use the equation for impulse, Impulse = F_avg x Δt, where Δt = time duration (0.10 s).
-2.91 kg·m/s = F_avg x 0.10 s
F_avg = -2.91 kg·m/s / 0.10 s
F_avg ≈ -29.1 N
The negative sign indicates that the force is exerted in the opposite direction of the ball's motion, which is typical for a decelerating force.
The average density of the body of a fish is 1080 kg/m3. To keep from sinking, the fish increases its volumeby inflating an internal air bladder. By what percentage must the fish increase its volume in order to beneutrally buoyant in fresh water? The density of air at room temperature is 1.2 kg/m3.
Answer:
f = 8 %
Explanation:
given,
density of body of fish = 1080 kg/m³
density of air = 1.2 Kg/m³
density of water = 1000 kg/m²
to protect the fish from sinking volume should increased by the factor f
density of fish + density of water x increase factor = volume changes in water
1080 +f x 1.2 =(1 + f ) x 1000
1080 + f x 1.2 = 1000 + 1000 f
998.8 f = 80
f = 0.0800
f = 8 %
the volume increase factor of fish will be equal to f = 8 %
In a double-star system, two stars of mass 4.6 x 1030 kg each rotate about the system's center of mass at radius 1.9 x 1011 m.
(a) What is their common angular speed?
(b) If a meteoroid passes through the system's center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system?
To carry out this exercise, it is necessary to use the equations made to Centripetal Force and Gravitational Energy Conservation.
By definition we know that the Centripetal Force is estimated as
[tex]F_c = M\omega^2R[/tex]
Where,
M = mass
[tex]\omega =[/tex] Angular velocity
R = Radius
From the 'linear' point of view the centripetal force can also be defined as
[tex]F_c = \frac{GM^2}{R^2}[/tex]
PART A ) Equating both equations we have,
[tex]\frac{M}{\omega^2R}=\frac{GM^2}{R^2}[/tex]
Re-arrange to find \omega
[tex]\omega = \sqrt{\frac{Gm}{r^3}}[/tex]
Replacing with our values
[tex]\omega = \sqrt{\frac{(6.67*10^{-11})(4.6*10^{30})}{(1.9*10^{11})^3}}[/tex]
[tex]\omega = 2.115*!0^{-7}rad/s[/tex]
Therefore the angular speed is [tex]\omega = 2.115*!0^{-7}rad/s[/tex]
PART B) For energy conservation we have to
[tex]KE_{min} = PE_{cm}[/tex]
Where,
[tex]KE_{min} =[/tex] Minimus Kinetic Energy
[tex]PE_{cm} =[/tex] Gravitational potential energy at the center of mass
Then,
[tex]\frac{1}{2} mv^2_{min} = \frac{2GMm}{R}[/tex]
Re-arrange to find v,
[tex]v_min = \sqrt{\frac{4GM}{R}}[/tex]
[tex]v_min = \sqrt{\frac{4(6.67*10^{-11})(2.2*10^{30})}{(1.9*10^{11})}}[/tex]
[tex]v_min = 5.55*10^4m/s[/tex]
Therefore the minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system is [tex]v_min = 5.55*10^4m/s[/tex]
Three cars (car L, car M, and car N) are moving with the same speed and slam on their brakes. The most massive car is car L, and the least massive is car N. If the tires of all three cars have identical coefficients of kinetic friction with the road surface, for which car is the amount of work done by friction in stopping it the greatest?A) The amount of work done by friction is the same for all cars.B) Car NC) Car MD) Car L
Answer:
I believe it's B
The amount of work done by friction in stopping a car is determined by the force of friction and the distance over which it acts. Car L will require the greatest force of friction to stop it, making the amount of work done by friction the greatest for Car L.
Explanation:The amount of work done by friction in stopping a car is determined by the force of friction and the distance over which it acts. Since all three cars have identical coefficients of kinetic friction with the road surface, the force of friction will be the same for all three cars. However, the stopping distance for each car will vary based on its mass.
Car L has the greatest mass, so it will require the greatest force of friction to stop it. Therefore, the amount of work done by friction in stopping car L will be the greatest among the three cars.
Car M and Car N will require less force of friction compared to car L, so the amount of work done by friction in stopping them will be less than that of Car L, but equal to each other.
A student decides to move a box of books into her dormitory room by pulling on a rope attached to the box. She pulls with a force of 170 N at an angle of 25◦ above the horizontal. The box has a mass of 27.4 kg, and the coefficient of friction between box and floor is 0.293. The acceleration of gravity is 9.8 m/s 2 . 27.4 kg µ = 0.293 170 N 25◦ Find the acceleration of the box. Answer in units of m/s 2 .
Answer:
a = 3.52 m/s²
Explanation:
Newton's second law:
∑F = m*a Formula (1)
∑F : algebraic sum of the forces in Newton (N)
m : mass s (kg)
a : acceleration (m/s²)
Data
m= 27.4 kg : mas of the box
F= 170 N, at an angle of 25◦ above the horizontal :Force rope attached to the box
μk = 0.293 :Coefficient of friction between box and floor
g = 9.8 m/s² : acceleration due to gravity
We define the x-axis in the direction parallel to the movement of the box and the y-axis in the direction perpendicular to it.
Forces acting on the box
W: Weight of the block : In vertical direction ,downward
FN : Normal force : perpendicular to the floor upward
f : Friction force: parallel to the floor and opposite to the movement
F : force of the rope attached to the box , at an angle of 25◦ above the horizontal
Calculated of the W ( weight of the box)
W= m*g
W= 27.4 kg* 9.8 m/s² = 268.52 N
x-y components of the force of 170 N
Fx=170 N *cos 25° = 154.07 N
Fy=170 N *sin 25° =71.845 N
Calculated of the FN ( Normal force)
We apply the formula (1)
∑Fy = m*ay ay = 0
FN + Fy - W = 0
FN = W- Fy
FN = 268.52 N - 71.845 N
FN =196. 675 N
Calculated of the f (friction force)
f = μk*FN
f = 0.293*196. 675
f = 57.626 N
We apply the formula (1) to calculated acceleration of the box:
∑Fx = m*ax , ax= a : acceleration of the box
Fx-f = m*a
154.07-57.626 = (27.4)*a
96.45 = (27.4)*a
a = (96.45)/ (27.4)
a = 3.52 m/s²
Final answer:
To calculate the box's acceleration, resolve the pulling force into components, find the normal and frictional forces, and apply Newton's second law with the net horizontal force and box's mass.
Explanation:
To find the acceleration of the box, we first need to resolve the applied force into horizontal and vertical components. The horizontal component of the force ( extit{F}_{horizontal}) is calculated by 170 N imes extit{cos}(25°), and the vertical component ( extit{F}_{vertical}) by 170 N imes extit{sin}(25°). The normal force ( extit{N}) is the sum of the vertical component and the weight of the box acting upwards, which is the box mass (27.4 kg) multiplied by the acceleration of gravity (9.8 m/s²), but since the box doesn't accelerate vertically, the normal force is equal to the weight minus extit{F}_{vertical}.
Next, we calculate the frictional force ( extit{F}_{friction}) using the coefficient of friction (0.293) and the normal force. The net horizontal force ( extit{F}_{net}) is the difference between the horizontal component of the pull and the frictional force. Finally, we use Newton's second law ( extit{F} = m imes extit{a}) to find the acceleration by dividing the extit{F}_{net} by the mass of the box (27.4 kg).
Which one of the following statements concerning the electric dipole moment is false?
A) The dipole moment is a scalar quantity.
B) The dipole moment has units of C·m.
C) The dipole moment combines two intrinsic properties of an electric dipole, the distance between the charges and the amount of charge on each object.
D) The dipole moment is directed from the negative charge toward the positive charge of the dipole.
E) The direction of the dipole moment is used to specify the orientation of the dipole.
Answer:
A.The dipole moment is a scalar quantity.
Explanation:
We know that
Dipole moment :It measure the polarity of chemical bond in a molecule .
The electric dipole moment is equal to product of any charge (positive or negative ) and the distance between the two charges.
Mathematical representation:
[tex]\mu=q\times d[/tex]
Where [tex]\mu[/tex] =Dipole moment
q=Charge on atom or particle
d=Distance between two charged particles
It helps to find out the molecule is polar or non- polar.
When the dipole moment is zero then the molecule is non-polar.
Dipole moment is a vector quantity.
The direction of dipole moment from negative charge to positive.
When unit of charge is C and unit of distance is m.
Then, unit of dipole moment=C-m
Hence, option A is false.
Final answer:
The false statement about the electric dipole moment is that it is a scalar quantity; it is a vector quantity. So the correct option is A.
Explanation:
The statement concerning the electric dipole moment that is false is A) The dipole moment is a scalar quantity. This is incorrect because the electric dipole moment is a vector quantity, not a scalar. The correct statements about the dipole moment are that it does indeed have units of C·m (Statement B), it combines both the distance between the charges and the magnitude of the charges (Statement C), it is directed from the negative charge toward the positive charge (Statement D), and its direction specifies the orientation of the dipole (Statement E).
Although the evidence is weak, there has been concern in recent years over possible health effects from the magnetic fields generated by electric transmission lines. A typical high-voltage transmission line is 20 m above the ground and carries a 200 A current at a potential of 110 kV.What is the magnetic field strength on the ground directly under such a transmission line?
The concept necessary to develop this exercise is that of a magnetic field on a surface. The magnetic field is a tool to describe how a magnetic force is distributed in the space around and within something magnetic.
It can be defined as,
[tex]B = \frac{\mu_0 I}{2\pi R}[/tex]
Where,
R= The distance from the point
I = Current
[tex]\mu_0 =[/tex] Permeability constant in free space
Our values are given as,
[tex]\mu_0 = 4\pi * 10^{-7}H/m[/tex]
[tex]I = 200 A[/tex]
[tex]R = 20m[/tex]
Replacing ,
[tex]B = \frac{(4\pi*10^{-7})(200)}{2\pi 20}[/tex]
[tex]B = 2*10^{-6}T[/tex]
Therefore the magnetic field strength on the ground directly under such a transmission line is [tex]2*10^{-6}T[/tex]
The magnetic field strength on the ground directly under a typical high-voltage transmission line carrying a 200 A current at a potential of 110 kV, which is 20 m above the ground, is 2 μT. This is calculated using Ampere's Law. Current evidence does not conclusively support any health hazards associated with such exposure.
Explanation:To answer your question about the strength of the magnetic field on the ground directly under a typical high-voltage transmission line carrying a 200 A current at a potential of 110 kV, we should first understand the basic physics principle. The magnetic field created by an electric current in a long straight wire forms concentric circles around the wire, and the strength B of the magnetic field is given by Ampere's law.
Ampere's law states that B = μ₀I / 2πr, where B is the magnetic field strength, μ₀ is the permeability of free space (4π × 10^-7 T m/A), I is the current, and r is the distance from the wire. In this case, the transmission line is 20 m above the ground, so r = 20 m. The current I = 200 A. Substituting these values into the equation, we get B = (4π × 10^-7 T m/A * 200 A) / (2π * 20 m) = 2 × 10^-6 T or 2 μT.
So, the strength of the magnetic field on the ground directly under the transmission line is 2 μT, which is less than a tenth of Earth's admittedly weak magnetic field. While there's an ongoing controversy regarding potential health hazards associated with exposure to these electromagnetic fields (E-fields), current evidence does not conclusively support these claims.
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An ideal parallel-plate capacitor consists of a set of two parallel plates of area Separated by a very small distance 푑. This capacitor is connected to a battery that charges the capacitor such that the energy stored in the capacitor is 푈'. Now the battery is disconnected and the separation between the plates is doubled, how much energy is stored in the capacitor?
Answer:
doubled the initial value
Explanation:
Let the area of plates be A and the separation between them is d.
Let V be the potential difference of the battery.
The energy stored in the capacitor is given by
U = Q^2/2C ...(1)
Now the battery is disconnected, it means the charge is constant.
the separation between the plates is doubled.
The capacitance of the parallel plate capacitor is inversely proportional to the distance between the plates.
C' = C/2
the new energy stored
U' = Q^2 / 2C'
U' = Q^2/C = 2 U
The energy stored in the capacitor is doubled the initial amount.
On a vacation flight, you look out the window of the jet and wonder about the forces exerted on the window. Suppose the air outside the window moves with a speed of approximately 150m/s shortly after takeoff, and that the air inside the plane is at atmospheric pressure.(A) Find the pressure difference between the inside and outside of the window.(B) If the window is 25cm by 45cm, find the force exerted on the window by air pressure.
The pressure difference between the inside and outside of the window can be found using Bernoulli's principle. The force exerted on the window by air pressure can be calculated using the formula F = P * A.
Explanation:A) The pressure difference between the inside and outside of the window can be found using Bernoulli's principle. According to Bernoulli's principle, the pressure difference is equal to the difference in kinetic energy per unit volume between the inside and outside air. This can be expressed as: ΔP = 1/2 ρv², where ΔP is the pressure difference, ρ is the density of air, and v is the speed of air. Substituting the given values, we get: ΔP = 1/2 * 1.14 kg/m³ * (150 m/s)² = 12,825 Pa.
B) The force exerted on the window by air pressure can be calculated using the formula: F = P * A, where F is the force, P is the pressure difference, and A is the area of the window. Substituting the given values, we get: F = 12,825 Pa * (0.25 m * 0.45 m) = 1,826.25 N.
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An angry rhino with a mass of 2600 kg charges directly toward you with a speed of 3.70 m/s. Before you start running, as a distraction, you throw a 0.180-kg rubber ball directly at the rhino with a speed of 8.39 m/s. Determine the speed of the ball after it bounces back elastically toward you.
Answer:
vf₂ = 15.79 m/s
Explanation:
Theory of collisions
Linear momentum is a vector magnitude (same direction of the velocity) and its magnitude is calculated like this:
p=m*v
where
p:Linear momentum
m: mass
v:velocity
There are 3 cases of collisions : elastic, inelastic and plastic.
For the three cases the total linear momentum quantity is conserved:
P₀ = Pf Formula (1)
P₀ :Initial linear momentum quantity
Pf : Final linear momentum quantity
Data
m₁ = 2600 kg : mass of the rhino
m₂= 0.18 kg : mass of the ball
v₀₁ = 3.70 m/s : initial velocity of the rhino
v₀₂= - 8.39 m/s, initial velocity of the ball
Problem development
We appy the formula (1):
P₀ = Pf
m₁*v₀₁ + m₂*v₀₂ = m₁*vf₁ + m₂*vf₂
(2600)*(3.7) + (0.18)*(- 8.39) = (2600)*vf₁ +(0.18)*vf₂
9620 -1.5102 = (2600)*vf₁ +(0.18)*vf₂
9618.4898 = (2600)*vf₁ +(0.18)*vf₂ Equation (1)
Because the shock is elastic, the coefficient of elastic restitution (e) is equal to 1.
[tex]e = \frac{v_{f2}-v_{f1} }{v_{o1}-v_{o2}}[/tex]
1*( v₀₁- v₀₂ ) = (vf₂ -vf₁)
( 3.7 -( -8.39 ) = (vf₂ -vf₁)
12.09 = (vf₂ -vf₁)
vf₂ = vf₁ + 12.09 Equation (2)
We replace Equation (2) in the Equation (1)
9618.4898 = (2600)*vf₁ +(0.18)*vf₂
9618.4898 = (2600)*vf₁ +(0.18)* (vf₁ + 12.09)
9618.4898 = (2600)*vf₁ +(0.18)*vf₁ + (0.18)(12.09)
9618.4898 = (2600)*vf₁ +(0.18)*vf₁ + 2.1762
9618.4898 -2.1762 = (2600.18)*vf₁
9616.3136 = (2600.18)*vf₁
vf₁ = (9616.3136) / (2600.18)
vf₁ = 3.698 m/s : Final velocity of the rhino
We replace vf₁ = 3.698 m/s in the Equation (2)
vf₂ = vf₁ + 12.09
vf₂ = 3.698 + 12.09
vf₂ = 15.79 m/s : Final velocity of the ball
The speed of the ball after it bounces back elastically toward you is 8.39 m/s.
To solve this problem, we can use the principle of conservation of momentum and the fact that the collision is elastic. In an elastic collision, both momentum and kinetic energy are conserved.
Using the conservation of momentum:
[tex]\[ m_r v_{r,i} + m_b v_{b,i} = m_r v_{r,f} + m_b v_{b,f} \][/tex]
Since the rhino is much more massive than the ball, its velocity will change very little after the collision. This assumption is valid in the limit where the mass of one object is much larger than the other, which is the case here.
With this assumption, the conservation of momentum simplifies to:
[tex]\[ m_b v_{b,i} = m_b v_{b,f} \][/tex]
Since the mass of the ball cancels out, we find that:
[tex]\[ v_{b,f} = v_{b,i} \][/tex]
This means that the speed of the ball after the collision is the same as before the collision, which is 8.39 m/s, but in the opposite direction.
Additionally, in an elastic collision, the relative speed of approach is equal to the relative speed of separation. Since the rhino's speed is much greater than the ball's speed, the ball will essentially rebound with the same speed it had before the collision but in the opposite direction.
A space shuttle in orbit around the Earth carries its payload with its mechanical arm. Suddenly, the arm malfunctions and releases the payload. What will happen to the payload?
a. It will fall straight down and hit the Earth.
b. It will follow a curved path and eventually hit the Earth.
c. It will drift out into deep space.
d. It will remain in the same orbit with the shuttle.
We must remember that a body in Earth orbit is under perpetual free fall, that is, that the body moves under a constant speed in constant around the earth.
The moment one of the charges is released, its displacement speed will be equal to that of the ship, so it will tend to continue rotating in the orbit in which the ship was traveling.
The correct answer is D. The payload will remain in the same orbit with the space shuttle.
(a) Steve exerts a steady force of magnitude 210N (about 47 lb.) on the stalled car in fig. 6.3 as he pushesit a distance of 18 m. The care also has a flat tire, so to make the car track straight Steve must push at an angle of 30 degrees to the direction of motion. How much work does Steve do? (b) refer to book.
Answer:
3 300 J
Explanation:
Work done, mathematically speaking, is the distance moved multiplied by the force exerted in the only in direction of motion. In this question therefore we will multiply the distance by the component of the force that moved the car forward
W = Fdcosθ
= 210 N * 18 * Cos 30°
= 3 300 J
The work done by Steve while pushing the car 18 meters at an angle of 30 degrees, applying a force of 210N, is calculated to be 3465 joules using the formula: Work = Force x distance x cos(angle).
Explanation:The subject of this question is Physics, specifically the concept of work done by a force. In physics, work is defined as the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
For Steve's situation, we can calculate the work done using the formula: Work = Force x distance x cos(angle). Given the parameters: Force = 210N, Distance = 18m, and Angle = 30 degrees, we get:
Work = 210N x 18m x cos(30) = 3465 J
So, Steve does 3465 Joules of work to push the car 18 meters under these conditions. This example illustrates how the angle at which force is applied can affect the work output, a key principle in physics.
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A 2-kg wood block is pulled by a string across a rough horizontal floor. The string exerts a tension force of 30 N on the block at an angle of 20º above the horizontal. The block moves at constant speed. If the block is pulled for a distance of 3.0 m, how much work is done by the tension force?
Answer:
Work done, W = 84.57 Joules
Explanation:
It is given that,
Mass of the wooden block, m = 2 kg
Tension force acting on the string, F = 30 N
Angle made by the block with the horizontal, [tex]\theta=20^{\circ}[/tex]
Distance covered by the block, d = 3 m
Let W is the work done by the tension force. It can be calculated as :
[tex]W=F\ cos\theta\times d[/tex]
[tex]W=30\times cos(20)\times 3[/tex]
W = 84.57 Joules
So, the work done by the tension force is 84.57 Joules. Hence, this is the required solution.
The quantity of work done by this tension force is equal to 84.57 Joules.
Given the following data:
Mass of wooden block, m = 2 kg
Tension force, F = 30 N
Angle = 20°
Distance, d = 3 m.
How to calculate the quantity of work done?In Science, work done is generally calculated by multiplying tension force and the vertical distance experienced by an object.
Mathematically, this is given by:
W = Fdcosθ
W = 30 × 3.0 × cos20
Work done, W = 84.57 Joules.
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The average density of the body of a fish is 1080kg/m^3 . To keep from sinking, the fish increases its volume by inflating an internal air bladder, known as a swim bladder, with air. By what percent must the fish increase its volume to be neutrally buoyant in fresh water? Use 1.28kg/m^3 for the density of air at 20 degrees Celsius.
To solve this problem it is necessary to apply the concept related to density and its definition with respect to mass and volume.
Density can be expressed as
[tex]\rho = \frac{m}{V}[/tex]
Where,
m = mass
V = Volume
The average density of fish air system is equal to the neutral bouyancy when the water displaced by the fish air system can be expressed as
[tex]\rho_{b} = \frac{m_a+m_f}{V_a+V_f}[/tex]
Where,
[tex]m_a[/tex]= Mass of air
[tex]m_f =[/tex] Mass of fish
[tex]V_a[/tex]= Volume of air
[tex]V_f[/tex]= Volume of fish
As we know that m = \rho V we can replacing the equivalent value for the mass fro fish and air, then
[tex]\rho_{b} = \frac{m_a+m_f}{V_a+V_f}[/tex]
[tex]\rho_{b} = \frac{\rho_a V_a+\rho_f V_f}{V_a+V_f}[/tex]
[tex]\rho_{b}(V_a+V_f)=\rho_a V_a+\rho_f V_f[/tex]
[tex]\rho_{b} V_a+\rho_{b}V_f = \rho_a V_a+\rho_f V_f[/tex]
[tex]\rho_{b} V_a- \rho_a V_a=\rho_f V_f-\rho_{b}V_f[/tex]
[tex]V_a(\rho_{b}-\rho_a)=V_f(\rho_f-\rho_{b})[/tex]
[tex]\frac{V_a}{\rho_f}=\frac{(\rho_f-\rho_{b})}{(\rho_{b}-\rho_a)}[/tex]
Replacing with our values we have that
[tex]\frac{V_a}{\rho_f}=\frac{(1000-1080)}{(1.2-1000)}[/tex]
[tex]\frac{V_a}{\rho_f}=0.08[/tex]
[tex]\frac{V_a}{\rho_f}=8\%[/tex]
Therefore the percentage of Volume is increased by 8%
To achieve neutral buoyancy in fresh water, a fish with an initial density of 1080 kg/m^3 needs to increase its volume by 8%, utilizing its swim bladder.
Explanation:The solution to this problem involves understanding Archimedes' Principle and the concept of density. In order to achieve neutral buoyancy, the fish needs to match its density to the density of freshwater, which is approximately 1000 kg/m³. It can do this by increasing its volume through inflating its swim bladder, thus reducing its overall density.
Since the average density of an object is mass divided by its volume, we can set up an equation involving the initial and final densities of the fish, with the initial density being 1080 kg/m³ and the final density being 1000 kg/m³. Let the initial volume of the fish be V1, the final volume be V2, and the mass of the fish (which doesn't change) be m. Therefore, m/V1 = 1080 and m/V2 = 1000. We can then solve these equations to find that V2 = 1.08V1.
In terms of percent increase, (V2 - V1) / V1 = 0.08 or 8%. Hence, the fish needs to increase its volume by 8% to achieve neutral buoyancy in freshwater.
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A 0.750-kg object hanging from a vertical spring is observed to oscillate with a period of 1.50 s. When the 0.750-kg object is removed and replaced by a 1.50-kg object, what will be the period of oscillation
Answer:
New time period, [tex]T_2=2.12\ s[/tex]
Explanation:
Given that,
Mass of the object 1, [tex]m_1=0.75\ kg[/tex]
Time period, [tex]T_1=1.5\ s[/tex]
If object 1 is replaced by object 2, [tex]m_2=1.5\ kg[/tex]
Let [tex]T_2[/tex] is the new period of oscillation.
The time period of oscillation of mass 1 is given by :
[tex]T_1=2\pi \sqrt{\dfrac{m_1}{k}}[/tex]
[tex]1.5=2\pi \sqrt{\dfrac{0.75}{k}}[/tex]............(1)
The time period of oscillation of mass 2 is given by :
[tex]T_2=2\pi \sqrt{\dfrac{m_2}{k}}[/tex]
[tex]T_2=2\pi \sqrt{\dfrac{1.5}{k}}[/tex]............(2)
From equation (1) and (2) we get :
[tex](\dfrac{T_1}{T_2})^2=\dfrac{m_1}{m_2}[/tex]
[tex](\dfrac{1.5}{T_2})^2=\dfrac{0.75}{1.5}[/tex]
[tex]\dfrac{1.5}{T_2}=0.707[/tex]
[tex]T_2=2.12\ s[/tex]
So, the new period of oscillation is 2.12 seconds. Hence, this is the required solution.
Final answer:
To find the new period of oscillation when the mass attached to a spring is increased, one can use the formula for the period T = 2π√(m/k). When the mass is doubled from 0.750 kg to 1.50 kg, the new period of oscillation is approximately 2.12 seconds.
Explanation:
The period of oscillation for a mass m suspended from a spring is given by the formula T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. Using this relationship, we can compare the periods for different masses to find the new period when the mass changes.
When the mass was 0.750 kg, the period was 1.50 s. Now, if the mass is increased to 1.50 kg, we can expect the period to change. Noting that the spring constant k remains the same, we use the ratio of the periods T1/T2 = √(m1/m2). Solving for the new period T2 gives us T2 = T1 * √(m2/m1).
If we plug in the numbers, we get:
T1 = 1.50 s (given for 0.750 kg)
m1 = 0.750 kg
m2 = 1.50 kg
T2 = 1.50 s * √(1.50 kg / 0.750 kg) = 1.50 s * √2 ≈ 2.12 s
The new period of oscillation for the 1.50-kg object is therefore approximately 2.12 seconds.
A fluid flows through two horizontal pipes of equal length which are connected together to form a pipe of length 2l. The flow is laminar and fully developed. The pressure drop for the first pipe is 1.789 times greater than it is for the second pipe. If the diameter of the first pipe is D, determine the diameter of the second pipe.
Answer:
Explanation:
For rate of flow of liquid in a pipe , the formula is
Q = π p r⁴ / 8 η l
where Q is volume of liquid flowing per unit time , p is pressure drop across pipe , r is radius of pipe , η is coefficient of viscosity and l is length of pipe
If η and l are constant for two pipes having equal rate of flow as in the given case
p₁ r₁⁴ = p₂ r₂⁴
p₁ = 1.789 p₂ ( given )
r₁ = D/2
1.789 p₂ x (D/2)⁴ = p₂ x r₂⁴
r₂ = [tex]\frac{D}{2}\times\sqrt[4 ]{1.789}[/tex]
2 x r₂ = D x 1.1565
D₂ = D x 1.1565
To determine the diameter of the second pipe, apply Poiseuille's law to relate the known pressure drop ratio to the radii of the pipes, and then solve for the radius, r², of the second pipe.
Explanation:The subject of this question pertains to the Physics of fluid mechanics, specifically dealing with the laminar flow of a fluid through pipes of different diameters and the associated pressure drops.
According to Poiseuille's law, which describes laminar flow in pipes, the pressure drop (ΔP) across a length of pipe is directly proportional to the length (L) and the viscosity (η) of the fluid, and inversely proportional to the fourth power of the radius (r) of the pipe.
Given that the pressure drop for the first pipe with diameter D is 1.789 times greater than the second pipe, we can establish the following relationship using Poiseuille's equation ΔP ∝ Lη/r^4 (ignoring constants for convenience since lengths are equal, and assuming the same fluid viscosity):
(ΔP1/ΔP2) = (r2⁴ / r1⁴)
1.789 = (r2⁴ / (D/2)⁴)
Therefore, solving for r2 (radius of the second pipe), we get:
r²= 0.5 * D * (1.789)^(1/4)
Let d2 be the diameter of the second pipe, then d2 = 2 * r2:
d2 = D * (1.789)^(1/4)
By calculating the fourth root of 1.789 and multiplying it by the diameter D of the first pipe, the diameter of the second pipe can be determined.
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When you take your 1900-kg car out for a spin, you go around a corner of radius 53m with a speed of 13m/s. The coefficient of static friction between the car and the road is 0.88. Assuming your car doesnt skid, what is the force exerted on it by static friction?
Answer:
Ff = 6058.5N
Explanation:
The sum of forces is:
[tex]Ff = m*a_c[/tex]
[tex]Ff = m*V^2/R[/tex]
[tex]Ff = 1900*13^2/53[/tex]
[tex]Ff = 6058.5N[/tex]
A 0.750kg block is attached to a spring with spring constant 13.5N/m . While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 32.0cm/s .
What are :
A) The amplitude of the subsequent oscillations?
Express your answer with the appropriate units.
B) The block's speed at the point where x= 0.750 A?
Express your answer with the appropriate units.
Answer:
A)A=0.075 m
B)v= 0.21 m/s
Explanation:
Given that
m = 0.75 kg
K= 13.5 N
The natural frequency of the block given as
[tex]\omega =\sqrt{\dfrac{K}{m}}[/tex]
The maximum speed v given as
[tex]v=\omega A[/tex]
A=Amplitude
[tex]v=\sqrt{\dfrac{K}{m}}\times A[/tex]
[tex]0.32=\sqrt{\dfrac{13.5}{0.75}}\times A[/tex]
A=0.075 m
A= 0.75 cm
The speed at distance x
[tex]v=\omega \sqrt{A^2-x^2}[/tex]
[tex]v=\sqrt{\dfrac{K}{m}}\times \sqrt{A^2-x^2}[/tex]
[tex]v=\sqrt{\dfrac{13.5}{0.75}}\times \sqrt{0.075^2-(0.075\times 0.75)^2}[/tex]
v= 0.21 m/s
The amplitude of the subsequent oscillations can be calculated using the formula for potential energy and the initial kinetic energy. The block's speed at a given point can be found using the conservation of mechanical energy and the equation for kinetic energy.
Explanation:To determine the amplitude of the subsequent oscillations, we need to first calculate the potential energy stored in the spring when the block is hit. Since the block is moving at a speed of 32.0 cm/s, we can calculate its kinetic energy. We can then equate this kinetic energy to the potential energy of the spring, using the formula potential energy = (1/2) * k * A^2, where k is the spring constant and A is the amplitude. Rearranging the formula, we find that the amplitude is equal to the square root of (2 * kinetic energy / k).
To find the block's speed when x = 0.750 A, we can use the conservation of mechanical energy. At the maximum displacement, all the energy is converted between potential and kinetic energy. At this point, the potential energy is zero, so the kinetic energy is equal to the initial kinetic energy. Using the formula for kinetic energy, we can find the speed when x = 0.750 A using the equation kinetic energy = (1/2) * k * (x^2 - A^2) and solving for the speed.
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A 0.72-m long string has a mass of 4.2 g. The string is under a tension of 84.1 N. What is the speed of a wave on this string?
The speed of the wave in the string is 83.4 m/s
Explanation:
For a standing wave in a string, the speed of the wave is given by the equation:
[tex]v=\frac{1}{2L}\sqrt{\frac{T}{m/L}}[/tex]
where
L is the length of the string
T is the tension in the string
m is the mass of the string
In this problem, we have:
L = 0.72 m
m = 4.2 g = 0.0042 kg
T = 84.1 N
Solving the equation, we find the speed of the wave:
[tex]v=\frac{1}{2(0.72)}\sqrt{\frac{84.1}{0.0042/0.72}}=83.4 m/s[/tex]
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If the radius of a blood vessel drops to 80.0% of its original radius because of the buildup of plaque, and the body responds by increasing the pressure difference across the blood vessel by 10.0%, what will have happened to the flow rate? The flow rate will have changed to _______ % of its original value.
Answer:
40.96%
Solution:
As per the question:
If the original radius be 'r' and the initial pressure difference be 'P':
After the drop, radius:
r' = 0.8r
After rise in pressure:
P' = 0.1 P
Now,
Rate of flow is given by:
R ∝ [tex]r^{4}P[/tex]
Thus
[tex]\frac{R'}{R} = (\frac{0.8r}{r})^{4}\frac{0.1P}{P}[/tex]
R' = 40.96%
Light of wavelength 600 nm in vacuum is incident nearly perpendicularly on a thin film whose index of refraction is 1.5. The light travels from the top surface of the film to the bottom surface, reflects from the bottom surface, and returns to the top surface, as the drawing indicates. What is the total (down-and-back) distance traveled by the light inside the film? Express your answer in terms of the wavelength λfilm of the light within the film.
Answer:
6λ_film
Explanation:
Light of wavelength λ 600 nm
Assuming length of thin film of water to be 1200 nm
Total distance traveled = 2×1200 ×10^{-9} = [tex]\frac{2.4\times10^-6}{4\times10^{-7} }[/tex] m
Now, Wavelength of light in film = λ/n
n= refractive index
= [tex]\frac{600\times10^{-9}}{1.5}[/tex] = 4×10^{-7}
No. of wavelengths = distance traveled/Wavelength of light in film
= [tex]\frac{2.4\times10^-6}{4\times10^{-7} }[/tex] = 6
therefore, the total (down-and-back) distance traveled by the light inside the film in terms of wavelength λfilm = 6λfilm
Final answer:
The total distance that the light travels inside the thin film is equal to the wavelength of light within the film (λfilm), as the light travels down and back up, covering a distance of 2t, which is equivalent to λfilm.
Explanation:
When light of wavelength 600 nm in vacuum (represented by λ) is incident on a thin film with an index of refraction of 1.5, we need to calculate the total distance traveled by the light within the film. The wavelength of light within the film (λfilm) is given by the formula λfilm = λ / n, where n is the index of refraction of the film. Given that light hits the film perpendicularly, it travels down to the bottom surface and back up, covering a distance of 2t, where t is the thickness of the film. To express the total distance in terms of λfilm, we use the relationship 2t = 2(λfilm / (2n)) = λfilm since n is already considered in the λfilm. Thus, the total distance traveled by the light inside the film is λfilm.
To get total internal reflection at the interface of water (refractive index 1.33) and a plastic whose refractive index is 1.54:
Which material must the light start in?
A. it doesn't matter
B. water
C. plastic
What is the critical angle?
_____ degrees
Answer:
Explanation:
Total internal reflection refers to a phenomenon that occurs when light travels from a denser medium to less dense medium, in which the incidence ray is inclined at an angle greater than a certain critical angle to the normal ( angle at which the refracted ray is equal to 90°), instead of refraction, the ray is reflected back into the material.
since it must start from a denser medium or a medium with higher refractive index, then it must start with plastics
to calculate critical angle,
we use Snell's law
n₁ sin C = n₂ sin r since r is 90° where n₁ =1.54, n₂ = 1.33
C = sin⁻¹ ( n₂ / n₁) = sin⁻¹ ( 1.33 / 1.54) = 59.7°
the critical angle = 59.7°
The light must start in the plastic for total internal reflection to occur. The critical angle, which is the angle of incidence where the refracted light is tangential to the boundary, is approximately 59.46 degrees.
Explanation:For total internal reflection to occur, the light must go from a material with a higher refractive index to one with a lower refractive index. In this case, the light must start in the plastic.
The critical angle is the angle of incidence for which the angle of refraction equals 90 degrees. It can be calculated using Snell's Law (n1*sin(θ1) = n2*sin(θ2)), where n1 and n2 are the refractive indices and θ1 and θ2 are the angles of incidence and refraction, respectively. If we set θ2 to 90 degrees and rearrange the equation we get: θ1 (critical angle) = arcsin(n2/n1).
Plugging in the given index values, we get: θ1 = arcsin(1.33/1.54), which is approximately 59.46 degrees.
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Earth turns on its axis about once every 24 hours. The Earth's equatorial radius is 6.38 x 106 m. If some catastrophe caused Earth to suddenly come to a screeching halt, with what speed would Earth's inhabitants who live at the equator go flying off Earth's surface?
To solve this problem it is necessary to apply the concepts related to the Period of a body and the relationship between angular velocity and linear velocity.
The angular velocity as a function of the period is described as
[tex]\omega = \frac{2\pi}{T}[/tex]
Where,
[tex]\omega =[/tex]Angular velocity
T = Period
At the same time the relationship between Angular velocity and linear velocity is described by the equation.
[tex]v = \omega r[/tex]
Where,
r = Radius
Our values are given as,
[tex]T = 24 hours[/tex]
[tex]T = 24hours (\frac{3600s}{1 hour})[/tex]
[tex]T = 86400s[/tex]
We also know that the radius of the earth (r) is approximately
[tex]6.38*10^6m[/tex]
Usando la ecuación de la velocidad angular entonces tenemos que
[tex]\omega = \frac{2\pi}{T}[/tex]
[tex]\omega = \frac{2\pi}{86400}[/tex]
[tex]\omega = 7.272*10^{-5}rad/s[/tex]
Then the linear velocity would be,
[tex]v = \omega *r[/tex]
x[tex]v = \omega *r[/tex]
[tex]v= 463.96m/s[/tex]
The speed would Earth's inhabitants who live at the equator go flying off Earth's surface is 463.96
You wish to boil 1.9 kg of water, which has a specific heat capacity of 4186 J/kg-K. The water is initially at room temperature (293 K). Water boils at 373 K.
1) How much energy must be added to the water by heating it in order for it to start boiling?
Answer:
6.36 105 J
Explanation:
In calorimetry all the energy given to a system is converted to heat and the equation for heat is
Q = m [tex]c_{e}[/tex] ΔT
The temperature can be in degrees Celsius or Kelvin since the interval between them is the same, substitute and calculate
Q = 1.9 4186 (373-293)
Q = 6.36 105 J
This heat equals the energy supplied
An object is at rest on top of a smooth sphere with a radius of ???? = 15.3 m that is buried exactly halfway under the ground. It then begins to slide down. At what height from the ground is the object no longer in contact with the sphere?
Answer:
10.2 m .
Explanation:
Let object falls by angle θ .
At any moment after the fall , there are two forces acting on the sphere
1 ) mg cosθ which is a component of weight towards the centre 2 ) normal reaction of the surface R .
mgcosθ - R is net force acting, which provides centripetal force
mgcosθ - R = mv² / r
But v² = 2g r( 1-cosθ ) [ object falls by height ( r - r cosθ ).
mgcosθ - R = m / r x 2g r( 1-cosθ )
When the object is no longer in touch with sphere,
R = 0
mgcosθ = m / r x 2g r( 1-cosθ )
3 gr cosθ = 2gr
cosθ = 2/3
height of fall
= r ( 1-cosθ )
r ( 1 - 2/3 )
1/3 r
1/3 x 15.3
5.1 m
Height from the ground
15.3 - 5.1
10.2 m .
The object sliding down the sphere loses contact at a point where its gravitational and centripetal forces become equal, corresponding to an angle θ where cosθ = 2/3. Given the sphere's radius of 15.3m, this corresponds to a height from the ground of 10.2 m.
Explanation:To solve this problem, we need to apply the concept of conservative forces and energy conservation. Initially, the object is at rest at the top of the sphere, thus it possesses gravitational potential energy and no kinetic energy. As the object slides down, its potential energy converts into kinetic energy until a point where the object no longer remains in contact with the sphere.
This occurs when the gravitational force acting towards the center of the sphere becomes equal to the centripetal force required for the object in circular motion. Mathematically, this correlates to the equation: mgcosθ = mv²/r.
By simplifying this equation, considering all the potential energy converts into kinetic energy at the point of leaving contact (mgh = 1/2mv²), and knowing the radius of the sphere, we can deduce that the object loses contact with the sphere at a point where the angle θ (from the vertical axis) is cosθ = 2/3. The height (h) it loses contact can then be determined by the relationship h = r(1 - cosθ). Given the radius r = 15.3m, we find the height from the ground to be 10.2 m.
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In a sample of 18-karat gold, 75 percent of the total mass is pure gold, while the rest is typically 16 percent silver and 9 percent copper. If the density of pure gold is rhogold=19.3g/cm3, while the densitites of silver and copper are respectively rhosilver=10.5g/cm3 and rhocopper=8.90g/cm3, what is the overall density rho18kt of this alloy of 18-karat gold?
Answer:
15.57 g/cm³
Explanation:
[tex]\rho_g[/tex] = Density of gold = 19.3 g/cm³
[tex]\rho_s[/tex] = Density of silver = 10.5 g/cm³
[tex]\rho_c[/tex] = Density of copper = 8.9 g/cm³
Assuming total mass as 1000 g
Volume of gold
[tex]V_g=\frac{0.75\times 1000}{19.3}\\\Rightarrow V_g=38.86\ cm^3[/tex]
Volume of silver
[tex]V_g=\frac{0.16\times 1000}{10.5}\\\Rightarrow V_g=15.238\ cm^3[/tex]
Volume of copper
[tex]V_c=\frac{0.09\times 1000}{8.9}\\\Rightarrow V_c=10.11\ cm^3[/tex]
Density of the alloy would be
[tex]\rho=\frac{M}{V_g+V_s+V_c}\\\Rightarrow \rho=\frac{1000}{38.86+15.238+10.11}\\\Rightarrow \rho=15.57\ g/cm^3[/tex]
The overall density of this alloy is 15.57 g/cm³
The overall density of an 18-karat gold alloy, which consists of 75% gold, 16% silver, and 9% copper, is approximately 15.2 grams per cubic centimeter given the known densities of the pure metals.
Explanation:The density of an alloy (a mixture of metals) is calculated by finding the volume-weighted average of the densities of the constituent metals. In the case of 18-karat gold, we have a mixture of 75% gold, 16% silver, and 9% copper. Using the given densities of these metals, we can calculate the overall density of the alloy as follows:
rho18kt = 0.75*rhogold+0.16*rhosilver+0.09*rhocopper = 0.75*19.3g/cm³+0.16*10.5g/cm³+0.09*8.90g/cm³ = 15.2g/cm³.
Therefore, the overall density of the 18-karat gold alloy, rho18kt, is approximately 15.2g/cm³.
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A construction worker uses a steel tape to measure the length of an aluminum support column. If the measured length is 17.700 m when the temperature is 21.2°C, what is the measured length when the temperature rises to 29.4°C? (Note: Don't neglect the expansion of the steel tape. Give your answer to three decimal places.)
The measured length when the temperature rises to 29.4°C is 17.712 m.
Explanation:To answer this question, we need to consider the thermal expansion of the steel tape and the aluminum support column. The coefficient of linear expansion for the steel tape is not given, so we will assume it to be the same as iron, which is approximately 12x10-6/°C.
We can use the formula:
ΔL = αL₀ΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L₀ is the initial length, and ΔT is the change in temperature.
Given that the initial length (L₀) is 17.700 m and the initial temperature is 21.2°C, we can calculate the change in length (ΔL) when the temperature rises to 29.4°C.
ΔL = (12x10-6)(17.700)(29.4 - 21.2) = 0.01226352 m
Therefore, the measured length when the temperature rises to 29.4°C is 17.700 + 0.01226352 = 17.712 m (rounded to three decimal places).
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The change in length of the aluminum column when the temperature rises can be found using the equation ∆L = αL0∆T, where ∆L is the change in length, α is the coefficient of linear expansion, L0 is the initial length, and ∆T is the change in temperature. Plugging in the values provided, we find the change in length is 3.25848 x 10-4 m, rounded to three decimal places.
Explanation:When a material undergoes thermal expansion, its length increases as the temperature rises. To calculate the change in length, we can use the equation ∆L = αL0∆T, where ∆L is the change in length, α is the coefficient of linear expansion, L0 is the initial length, and ∆T is the change in temperature. In this case, we need to find the change in length of the aluminum column when the temperature rises from 21.2°C to 29.4°C.
First, we need to determine the coefficient of linear expansion for aluminum. Using the information provided, we can find that the coefficient of linear expansion for aluminum is approximately 22·10-6 °C-1.
Next, we can plug in the values into the equation. Given L0 = 17.700 m, ∆T = (29.4 - 21.2)°C = 8.2°C, and α = 22·10-6 °C-1, we can calculate ∆L as follows:
∆L = (22·10-6 °C-1)(17.700 m)(8.2°C) = 3.25848 x 10-4 m
Rounded to three decimal places, the measured length of the aluminum support column when the temperature rises to 29.4°C is 0.000 m.
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