Answer:
It is necessary to know the radius of the planet.
Explanation:
The speed of the spacecraft can be found by means of the equation of the Universal law of gravity:
[tex]F = G \frac{M.m}{r^{2}}[/tex] (1)
Where F is the gravitational force, G is gravitational constant, M is the mass of the planet, m is the mass of the spacecraft and r is the orbital radius of the spacecraft.
Equation 1 can be express in terms of the speed by using Newton's second law and the equation for centripetal acceleration:
[tex]F = ma[/tex] (2)
Replacing equation 2 in equation 1 it is gotten:
[tex]ma = G \frac{M.m}{r^{2}}[/tex] (3)
the centripetal acceleration is defined as:
[tex]a = \frac{v^{2}}{r}[/tex] (4)
Replacing equation 4 in equation 3 it is gotten:
[tex]m\frac{v^{2}}{r} = G \frac{M.m}{r^{2}}[/tex] (5)
Then, v can be isolated from equation 5:
[tex]mv^{2} = G \frac{M.m}{r}[/tex]
[tex]v^{2} = G \frac{M.m}{rm}[/tex]
[tex]v^{2} = G \frac{M}{r}[/tex]
[tex]v = \sqrt{\frac{GM}{r}}[/tex]
However, the orbital radius of the spacecraft is obtained by the sum of the radius of the planet and the height of the spacecraft above the surface of the planet (r = R+h)
[tex]v = \sqrt{\frac{GM}{R+h}}[/tex] (6)
Hence, by equation 6 it can be concluded that it is necessary to know the radius of the planet in order to calculate the speed of the spacecraft.
A ray of light is incident on a flat surface of a block of polystyrene, with an index of refraction of 1.49, that is submerged in water. The ray is split at the surface, where the angle of refraction of the transmitted ray is 18.6°. What is the angle of reflection (in degrees) of the reflected ray?
To solve this problem it is necessary to apply the Snell Law. With which the angles of refraction and incidence on two materials with a determined index of refraction are described.
The equation stipulates that
[tex]n_1 sin\theta_1 = n_2 sin\theta_2[/tex]
Where,
[tex]n_i[/tex]= Index of refraction of each material
[tex]\theta_1 =[/tex] Angle of incidence or Angle of Reflection
[tex]\theta_2 =[/tex] Angle of refraction
Our values are given as,
[tex]n_1 = 1.33 \rightarrow[/tex] Index of refraction of water
[tex]n_2 = 1.49[/tex]
[tex]\theta = 18.6\°[/tex]
Replacing we have that,
[tex]n_1 sin\theta_1 = n_2 sin\theta_2[/tex]
[tex](1.33) sin\theta_1 = (1.49)sin18.6[/tex]
[tex]\theta_1 = sin^{-1} (\frac{(1.49)sin18.6}{1.33})[/tex]
[tex]\theta_1 = 20.93\°[/tex]
Therefore the angle of reflection is 20.93°
A thin coil has 17 rectangular turns of wire. When a current of 4 A runs through the coil, there is a total flux of 5 ✕ 10−3 T · m2 enclosed by one turn of the coil (note that
Φ = kI,
and you can calculate the proportionality constant k). Determine the inductance in henries.
Answer:
Inductance, L = 0.0212 Henries
Explanation:
It is given that,
Number of turns, N = 17
Current through the coil, I = 4 A
The total flux enclosed by the one turn of the coil, [tex]\phi=5\times 10^{-3}\ Tm^2[/tex]
The relation between the self inductance and the magnetic flux is given by :
[tex]L=\dfrac{N\phi}{I}[/tex]
[tex]L=\dfrac{17\times 5\times 10^{-3}}{4}[/tex]
L = 0.0212 Henries
So, the inductance of the coil is 0.0212 Henries. Hence, this is the required solution.
The inductance of a coil with 17 turns, which has a flux of 5 * 10^−3 T · m2 when a current of 4 A runs through it, is 0.02125 Henry.
Explanation:In this problem, we are given the total flux Φ which is equal to the product of a proportionality constant k and the current I (Φ = kI ). The proportionality constant k can be calculated by dividing the flux by the current. k = Φ/I = (5 * 10^−3 T · m2) / 4 A = 1.25 * 10^-3 H/A. However, this is the inductance for just one single turn of the coil.
Since the coil has 17 turns, the total inductance L for the entire coil is equal to the product of the proportionality constant k and the number of turns N (L = kN). Therefore, L = 1.25 * 10^-3 H/A * 17 = 0.02125 Henry (H).
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The angular momentum about the center of the planet and the total mechanical energy will be conserved regardless of whether the object moves from small R to large R (like a rocket being launched) or from large R to small R (like a comet approaching the earth).
O True O False
Answer:
True
Explanation:
The angular momentum around the center of the planet and the total mechanical energy will be preserved irrespective of whether the object moves from large R to small R. But on the other hand the kinetic energy of the planet will not be conserved because it can change from kinetic energy to potential energy.
Therefore the given statement is True.
The correct answer is option A. The statement is true because in celestial mechanics, angular momentum and total energy are conserved under gravity, regardless of whether an object moves from a smaller to a larger radius or vice versa.
Let's explain these concepts:
1. Conservation of Angular Momentum (L):
The angular momentum of an object with respect to the center of the planet is given by the product of its linear momentum (p) and the radius (r) of its orbit:
[tex]\[ L = p \times r = mvr \][/tex]
where m is the mass of the object, v is its velocity, and r is the radius of the orbit. For a central force like gravity, the torque on the object is zero, which means that the angular momentum is conserved.
2. Conservation of Total Mechanical Energy (E):
The total mechanical energy of an object in orbit is the sum of its kinetic energy (K) and potential energy (U):
[tex]\[ E = K + U = \frac{1}{2}mv^2 - \frac{GMm}{r} \][/tex]
where G is the gravitational constant, M is the mass of the planet, and m is the mass of the object.
The complete question is:
The angular momentum about the center of the planet and the total mechanical energy will be conserved regardless of whether the object moves from small R to large R (like a rocket being launched) or from large R to small R (like a comet approaching the earth).
A. True
B. False
A rectangular loop of area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to B = Bmax e-t/τ , where Bmax and τ are constants. The field has the constant value Bmax for t < 0. Find the emf induced in the loop as a function of time. (Use the following as necessary: A, Bmax, t, and τ.) g
Answer:
Induced emf, [tex]\epsilon=-A\dfrac{B_{max}e^{-t/\tau}}{\tau}[/tex]
Explanation:
The varying magnetic field with time t is given by according to equation as :
[tex]B=B_{max}e^{-t/\tau}[/tex]
Where
[tex]B_{max}\ and\ t[/tex] are constant
Let [tex]\epsilon[/tex] is the emf induced in the loop as a function of time. We know that the rate of change of magnetic flux is equal to the induced emf as:
[tex]\epsilon=-\dfrac{d\phi}{dt}[/tex]
[tex]\epsilon=-\dfrac{d(BA)}{dt}[/tex]
[tex]\epsilon=-A\dfrac{d(B)}{dt}[/tex]
[tex]\epsilon=-A\dfrac{d(B_{max}e^{-t/\tau})}{dt}[/tex]
[tex]\epsilon=A\dfrac{B_{max}e^{-t/\tau}}{\tau}[/tex]
So, the induced emf in the loop as a function of time is [tex]A\dfrac{B_{max}e^{-t/\tau}}{\tau}[/tex]. Hence, this is the required solution.
The emf induced in the loop as a function of time is determined as [tex]emf = \frac{AB_{max}}{\tau} e^{-t/\tau}[/tex].
Induced emf
The emf induced in the rectangular loop is determined by applying Faraday's law of electromagnetic induction.
emf = dФ/dt
where;
Ф is magnetic fluxФ = BA
A is the area of the rectangular loop
emf = d(BA)/dt
[tex]emf = \frac{d(BA)}{dt} \\\\emf = A \frac{dB}{dt} \\\\emf = A \times B_{max}(e^{-t/\tau})(1/\tau)\\\\emf = \frac{A B_{max}(e^{-t/\tau})}{\tau}\\\\emf = \frac{AB_{max}}{\tau} e^{-t/\tau}[/tex]
Thus, the emf induced in the loop as a function of time is determined as [tex]emf = \frac{AB_{max}}{\tau} e^{-t/\tau}[/tex].
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A 2.10 kg textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is 0.150 m, to a hanging book with mass 3.20 kg. The system is released from rest, and the books are observed to move 1.20 m in 0.900 s.
(a) What is the tension in each part of the cord?
(b) What is the moment of inertia of the pulley about its rotation axis? Can you explain why the tensions are not the same even though it is the same cord.
The tension in the cord is different on both sides of the pulley due to its moment of inertia. The tensions can be calculated using the equation of motion and Newton's second law, while the moment of inertia needs the pulley's mass, which is not given.
Explanation:In a typical pulley system like the one described in the question, the force applied to the system is the force of gravity acting on the hanging mass. The tension in the cord is different on both sides due to the moment of inertia of the pulley, which resists rotation. This is why the tensions are not equal.
To find the tension (T1 and T2), we first need to calculate the acceleration (a) of the system using the equation for motion: a = 2d/t^2 = 2*1.2/0.9^2 = 2.96 m/s^2. We can then use Newton’s second law to express T1 as T1=F - m1*a = m1*g - m1*a = 2.10 kg * (9.81 - 2.96) m/s^2 = 14.38 N. Similarly, we can find T2 as T2=m2*a =3.20 kg * 2.96 m/s^2 = 9.47 N.
As for the moment of inertia of the pulley, its equation (when it is a solid cylinder) is I = 0.5*m*r^2, where m is the pulley’s mass and r its radius. Unfortunately, as we do not have the mass of the pulley in the information given, we cannot calculate the moment of inertia.
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To calculate the tension in the cord, one must apply Newton's second law to each mass, considering the pulley's moment of inertia influences the tensions. The pulley resists changes to its rotational motion due to its moment of inertia, leading to different tensions on either side.
Explanation:The question involves a physics problem related to Newton's laws, tension in a rope, and moment of inertia of a pulley. To calculate the tension in the cord for the given scenario, one must apply Newton's second law (F = ma) to each mass separately and then use the fact that the acceleration of both masses will be the same due to their connection by the rope. However, the tension will differ on either side of the pulley due to the pulley's moment of inertia, which affects how the cord transmits force.
For part (b), the moment of inertia of the pulley can be found by using the rotational analog of Newton's second law for the pulley and the fact that a torque applied to the pulley is equal to the difference in tension times the radius of the pulley. We must understand that the tensions differ because the pulley has a moment of inertia, which means it resists changes in its rotational motion, causing a difference in force on either side of the pulley.
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8. A baseball batter angularly accelerates a bat from rest to 20 rad/s in 40 ms (milliseconds). If the bat’s moment of inertia is 0.6 kg m2, then findA) the torque applied to the bat andB) the angle through which the bat moved.
Answer:
A) τ = 300 N*m
B) θ = 0.4 rad = 22.9°
Explanation:
Newton's second law:
F = ma has the equivalent for rotation:
τ = I * α Formula (1)
where:
τ : It is the moment applied to the body. (Nxm)
I : it is the moment of inertia of the body with respect to the axis of rotation (kg*m²)
α : It is angular acceleration. (rad/s²)
Data
I = 0.6 kg*m² : moment of inertia of the bat
Angular acceleration of the bat
We apply the equations of circular motion uniformly accelerated :
ωf= ω₀ + α*t Formula (2)
Where:
α : Angular acceleration (rad/s²)
ω₀ : Initial angular speed ( rad/s)
ωf : Final angular speed ( rad
t : time interval (s)
Data
ω₀ = 0
ωf = 20 rad/s
t = 40 ms = 0.04 s
We replace data in the formula (2) :
ωf= ω₀ + α*t
20 = 0 + α* (0.04)
α = 20/ (0.04)
α = 500 rad/s²
Newton's second law to the bat
τ = (0.6 kg*m²) *(500 rad/s²) = 300 (kg*m/s²)* m
τ = 300 N*m
B) Angle through which the bat moved.
We apply the equations of circular motion uniformly accelerated :
ωf²= ω₀ ²+ 2α*θ Formula (3)
Where:
θ : Angle that the body has rotated in a given time interval (rad)
We replace data in the formula (3):
ωf²= ω₀²+ 2α*θ
(20)²= (0)²+ 2(500 )*θ
400 = 1000*θ
θ = 400/1000
θ = 0.4 rad
π rad = 180°
θ = 0.4 rad *(180°/π rad)
θ = 22.9°
A solid 0.6350 kg ball rolls without slipping down a track toward a vertical loop of radius R=0.8950 m.
What minimum translational speed vmin must the ball have when it is a height H=1.329 m above the bottom of the loop in order to complete the loop without falling off the track?
Assume that the radius of the ball itself is much smaller than the loop radius R. Use g=9.810 m/s2 for the acceleration due to gravity.
The minimum translational speed the ball must have at a height of 1.329 m to complete the loop without falling off the track is approximately 6.61 m/s.
How can you solve the minimum translational speed the ball must have?
E p(top) = K e(bottom)
E p(top) = m * g * (R + H)
where:
m is the mass of the ball (0.6350 kg)
g is the acceleration due to gravity (9.810 m/s²)
R is the radius of the loop (0.8950 m)
H is the height above the bottom of the loop (1.329 m)
Calculate the minimum kinetic energy at the bottom:
Since the ball needs enough speed at the bottom to reach the top again, the minimum kinetic energy at the bottom is equal to the potential energy at the top:
K e(bottom) = E p(top) = m * g * (R + H)
Find the minimum translational speed:
K e = 1/2 * m * vmin²
where vmin is the minimum translational speed we're looking for. Solving for vmin:
v min = √(2 * K e / m) = √(2 * m * g * (R + H) / m)
v min = √(2 * g * (R + H))
Plug in the values and calculate:
v min = √(2 * 9.810 * (0.8950 + 1.329))
v min ≈ 6.61 m/s
Therefore, the minimum translational speed the ball must have at a height of 1.329 m to complete the loop without falling off the track is approximately 6.61 m/s.
An empty rubber balloon has a mass of 12.5 g. The balloon is filled with helium at a density of 0.181 kg/m3. At this density the balloon has a radius of 0.294 m. If the filled balloon is fastened to a vertical line, what is the tension in the line? The density of air is 1.29 kg/m3.
Answer: 1.14 N
Explanation :
As any body submerged in a fluid, it receives an upward force equal to the weight of the fluid removed by the body, which can be expressed as follows:
Fb = δair . Vb . g = 1.29 kg/m3 . 4/3 π (0.294)3 m3. 9.8 m/s2
Fb = 1.34 N
In the downward direction, we have 2 external forces acting upon the balloon: gravity and the tension in the line, which sum must be equal to the buoyant force, as the balloon is at rest.
We can get the gravity force as follows:
Fg = (mb +mhe) g
The mass of helium can be calculated as the product of the density of the helium times the volume of the balloon (assumed to be a perfect sphere), as follows:
MHe = δHe . 4/3 π (0.294)3 m3 = 0.019 kg
Fg = (0.012 kg + 0.019 kg) . 9.8 m/s2 = 0.2 N
Equating both sides of Newton´s 2nd Law in the vertical direction:
T + Fg = Fb
T = Fb – Fg = 1.34 N – 0.2 N = 1.14 N
When you apply the torque equation ∑τ = 0 to an object in equilibrium, the axis about which torques are calculated:
a. must be located at a pivot.
b. should be located at the edge of the object.
c. must be located at the object's center of gravity.
d. can be located anywhere.
Answer:
option D.
Explanation:
The correct answer is option D.
When an object is in equilibrium torque calculated at any point will be equal to zero.
An object is said to be in equilibrium net moment acting on the body should be equal to zero.
If the net moment on the object is not equal to zero then the object will rotate it will not be stable.
In a system in equilibrium, the choice of axis about which torques are calculated can be anywhere because for such a system, the sum of torques about any point is zero.
Explanation:When applying the torque equation, ∑τ = 0, to an object in equilibrium, the chosen axis about which torques are calculated can be located anywhere on or off the object. This principle is a result of the fact that in a system in equilibrium, the sum of torques about any point is zero, not just specific points like the object's pivot, edge, or center of gravity. For example, if you have a seesaw in balance, you could calculate the torques from the center, one of the seats, or even a point suspended above it in the air. As long as the object is in equilibrium and not moving, the torques calculated from any point will sum to zero because they counteract each other in direction and magnitude to maintain the state of balance.
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Two carts mounted on an air track are moving toward one another. Cart 1 has a speed of 2.20 m/s and a mass of 0.440 kg. Cart 2 has a mass of 0.740 kg. (a) If the total momentum of the system is to be zero, what is the initial speed of cart 2
Answer:
[tex]v_2=-1.308\ m/s[/tex]
Explanation:
It is given that,
Mass of cart 1, [tex]m_1=0.44\ kg[/tex]
Mass of cart 2, [tex]m_2=0.74\ kg[/tex]
Speed of cart 1, [tex]v_1=2.2\ m/s[/tex]
If the total momentum of the system is zero, the initial momentum is equal to the final momentum as :
[tex]m_1v_1=m_2v_2[/tex]
Let [tex]v_2[/tex] is the initial speed of cart 2
[tex]0.44\times 2.2=0.74v_2[/tex]
[tex]v_2=-1.308\ m/s[/tex]
Initial speed of the cart 2 is 1.308 m/s and it is moving in opposite direction of the cart 1.
As part of a safety investigation, two 1400 kg cars traveling at 20 m/s are crashed into different barriers. Find the average forces exerted on (a) the car that hits a line of water barrels and takes 1.5 s to stop, and (b) the car that hits a concrete barrier and takes 0.10 s to stop.
Answer:
(a) 18667N
(b)280000N
Explanation:
The momentum of both car prior to collision:
M = mv = 1400*20 = 28000 kgm/s
After colliding, the average force exerting on either car to kill this momentum is
(a)[tex] F_1 =\frac{M}{t_1} = \frac{28000}{1.5} = 18667N[/tex]
(b)[tex]F_2 = \frac{M}{t_2} = \frac{28000}{0.1} = 280000N[/tex]
To find the average forces exerted on the cars, use the formula: Force = change in momentum / time interval. For the car hitting the line of water barrels, the average force is -1400 kg * m/s / 1.5 s. For the car hitting the concrete barrier, the average force is -1400 kg * m/s / 0.10 s.
Explanation:To find the average forces exerted on the cars, we can use the formula:
Force = change in momentum / time interval
(a) For the car that hits a line of water barrels and takes 1.5 s to stop:
The change in momentum is given by: Δp = m * Δv = m * (0 - 20) = -1400 kg * m/s
Then, the average force exerted on the car is: F = Δp / t = -1400 kg * m/s / 1.5 s
(b) For the car that hits a concrete barrier and takes 0.10 s to stop:
The change in momentum is given by: Δp = m * Δv = m * (0 - 20) = -1400 kg * m/s
Then, the average force exerted on the car is: F = Δp / t = -1400 kg * m/s / 0.10 s
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A roundabout is a type of playground equipment involving a large flat metal disk that is able to spin about its center axis. A roundabout of mass 120kg has a radius of 1.0m is initially at rest. A child of mass 43kg is running toward the edge of the roundabout (meaning, running on a path tangent to the edge) at 2.7 m/s and jumps on. Once she jumps on the roundabout, they move together as a single object. Assume the roundabout is a uniform disk. 1. What is the magnitude of her angular momentum (with respect to the center of the roundabout) just before she jumps? 2. What is the angular speed of the roundabout after the jump? 3. Does the overall kinetic energy of the system increase, decrease, or remain constant? If you say it changed, explain what caused a change in energy.
Answer:
116.1 kgm²/s
1.12718 rad/s
Decreases
Explanation:
m = Mass of girl = 43 kg
M = Mass of roundabout = 120 kg
v = Velocity of roundabout = 2.7 m/s
r = Radius of roundabout = 1 m = R
I = Moment of inertia
Her angular momentum
[tex]L_i=mvr\\\Rightarrow L_i=43\times 2.7\times 1\\\Rightarrow L_i=116.1\ kgm^2/s[/tex]
Magnitude of angular momentum is 116.1 kgm²/s
Here the angular momentum is conserved
[tex]L_f=L_i\\\Rightarrow I\omega=L_i\\\Rightarrow (\frac{1}{2}MR^2+mr^2)\omega=116.1\\\Rightarrow \omega=\frac{116.1}{\frac{1}{2}\times 120\times 1^2+43\times 1^2}\\\Rightarrow \omega=1.12718\ rad/s[/tex]
Angular speed of the roundabout is 1.12718 rad/s
Initial kinetic energy
[tex]K_i=\frac{1}{2}mv^2\\\Rightarrow K_i=\frac{1}{2}43\times 2.7^2\\\Rightarrow K_i=156.735\ J[/tex]
Final kinetic energy
[tex]K_f=\frac{1}{2}I\omega^2\\\Rightarrow K_f=\frac{1}{2}\times (\frac{1}{2}\times 120\times 1^2+43\times 1^2)\times 1.12718^2\\\Rightarrow K_f=65.43253\ J[/tex]
The overall kinetic energy decreases as can be seen. This loss is converted to heat.
To find the angular momentum of the child just before she jumps onto the roundabout, consider her linear momentum and the moment of inertia of the roundabout. The angular speed of the roundabout after the jump can be found using the principle of conservation of angular momentum. The overall kinetic energy of the system remains constant.
Explanation:To find the angular momentum of the child just before she jumps onto the roundabout, we need to consider her linear momentum and the moment of inertia of the roundabout. The linear momentum of the child is given by the product of her mass and velocity. The angular momentum is then equal to the linear momentum multiplied by the distance from the center of the roundabout.
The angular speed of the roundabout after the jump can be found using the principle of conservation of angular momentum. The initial angular momentum of the system (just the roundabout) is zero, and since angular momentum is conserved, the final angular momentum after the child jumps on is equal to the angular momentum of the child just before she jumps.
The overall kinetic energy of the system remains constant. As the child jumps onto the roundabout, an external force (the ground pushing on the child) does work to change the linear momentum of the child, but no external torque acts on the system. So, the total mechanical energy (kinetic plus potential) is conserved.
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A 0.270 m radius, 510-turn coil is rotated one-fourth of a revolution in 4.17 ms, originally having its plane perpendicular to a uniform magnetic field. (This is 60 rev/s.) Find the magnetic field strength needed to induce an average emf of 10,000 V.
Answer:
0.35701 T
Explanation:
[tex]B_i[/tex] = Initial magnetic field
[tex]B_f[/tex] = Final magnetic field
[tex]\phi[/tex] = Magnetic flux
t = Time taken = 4.17 ms
N = Number of turns = 510
[tex]\epsilon[/tex] = Induced emf = 10000 V
r = Radius = 0.27 m
A = Area = [tex]\pi r^2[/tex]
Induced emf is given by
[tex]\epsilon=-N\frac{d\phi}{dt}\\\Rightarrow \epsilon=-N\frac{B_fAcos90-B_iAcos0}{dt}\\\Rightarrow \epsilon=N\frac{B_iA}{dt}\\\Rightarrow B_i=\frac{\epsilon dt}{NA}\\\Rightarrow B_i=\frac{10000 \times 4.17\times 10^{-3}}{510\times \pi 0.27^2}\\\Rightarrow B_i=0.35701\ T[/tex]
The magnetic field strength needed is 0.35701 T
A glass optical fiber is used to transport a light ray across a long distance. The fiber has an index of refraction of 1.550 and is submerged in ethyl alcohol, which has an index of refraction of 1.361. What is the critical angle (in degrees) for the light ray to remain inside the fiber?
To solve this exercise it is necessary to apply the concepts related to the Snells law.
The law defines that,
[tex]n_1 sin\theta_1 = n_2 sin\theta_2[/tex]
[tex]n_1 =[/tex] Incident index
[tex]n_2 =[/tex] Refracted index
[tex]\theta_1[/tex] = Incident angle
[tex]\theta_2 =[/tex] Refracted angle
Our values are given by
[tex]n_1 = 1.550[/tex]
[tex]n_2 = 1.361[/tex]
[tex]\theta_2 =90\° \rightarrow[/tex]Refractory angle generated when light passes through the fiber.
Replacing we have,
[tex](1.55)sin \theta_1 = (1.361) sin90[/tex]
[tex]sin \theta_1 = \frac{(1.361) sin90}{(1.55)}[/tex]
[tex]\theta_1 =sin^{-1} \frac{(1.361) sin90}{(1.55)}[/tex]
[tex]\theta_1 =61.4\°[/tex]
Now for the calculation of the maximum angle we will subtract the minimum value previously found at the angle of 90 degrees which is the maximum. Then,
[tex]\theta_{max} = 90-\theta \\\theta_{max} =90-61.4\\\theta_{max}=28.6\°[/tex]
Therefore the critical angle for the light ray to remain insider the fiber is 28.6°
If the molecular weight of air is 28.9, what is the density of air at atmospheric pressure and a temperature of 354.5 K? 1 atm = 1.013 × 105 N/m2 , the mass of a proton is 1.67262 × 10−27 kg , Avogadro’s number is 6.02214 × 1023 mol−1 and k = 1.38065 × 10−23 N · m/K . Answer in units of kg/m3
The density of air can be calculated at a given temperature and pressure using the ideal gas law. In order to get the result in kg/m³, we need to use the molecular weight of air and Avogadro's number to convert the density from moles per volume to kg/m³.
Explanation:To calculate the density of air at atmospheric pressure and a temperature of 354.5 K, we will be using the ideal gas equation which states that pV = nRT, where p is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. By rearranging the equation to solve for density (n/V), we can find that n/V = p/RT. We know the pressure (p) is 1.013 × 10⁵ N/m² (1 atm), the temperature (T) is 354.5 K, and the value of R (the Boltzmann constant) is 1.38065 × 10−23 N*m/K. Lastly, we need to use the molecular weight of air (28.9) and Avogadro's number to convert this into a density. Therefore, the calculation will be as follows:
density = (p*M)/(R*T) where M is the molar mass. Since we have all these values, we can substitute them into the equation.
In these calculations, it is important to convert the units appropriately to achieve the final answer in kg/m³.
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Using the ideal gas law, the density of air at a pressure of 1.013 × 10⁵ N/m² and a temperature of 354.5 K is calculated to be approximately 0.00971 kg/m³.
To determine the density of air at atmospheric pressure (1.013 × 10⁵ N/m²) and a temperature of 354.5 K given the molecular weight of air is 28.9, we can use the Ideal Gas Law:
PV = nRT
Rewriting for density (ρ = mass/volume) and using the molecular weight (M), the Ideal Gas Law can be expressed as:
ρ = (PM) / (RT)
Where:
P = Pressure = 1.013 × 10⁵ N/m²M = Molecular weight = 28.9 g/mol = 0.0289 kg/molR = Universal Gas Constant = 8.314 J/(mol.K)T = Temperature = 354.5 KSubstitute the values into the equation:
ρ = (1.013 × 10⁵ N/m² × 0.0289 kg/mol) / (8.314 J/(mol·K) × 354.5 K)
ρ ≈ 0.99 kg/m³
Thus, the density of air at the given conditions is approximately 0.99 kg/m³.
A large, simple pendulum is on display in the lobby of the United Nations building. If the pendulum is 18.5 m in length, what is the least amount of time it takes for the bob to swing from a position of maximum displacement to the equilibrium position of the pendulum? (Assume that the acceleration of gravity is g = 9.81 m/s2 at the UN building.)
Final answer:
The minimum time for a simple pendulum of 18.5 meters to swing from maximum displacement to equilibrium position is approximately 2.160 seconds, which is one-fourth the total period of the pendulum.
Explanation:
The minimum time for a pendulum to swing from maximum displacement to the equilibrium position is one-fourth of its period. The period of a simple pendulum, which is the time for one complete to-and-fro swing, can be calculated using the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Given a length, L, of 18.5 meters and gravity, g, of 9.81 m/s², we find:
T = 2π√(18.5/9.81) ≈ 2π√(1.886) ≈ 2π×1.373≈ 8.639 seconds
Therefore, the least amount of time it takes for the bob to swing from maximum displacement to the equilibrium position would be T/4:
T/4 = 8.639 seconds / 4 ≈ 2.160 seconds.
Final answer:
The least amount of time for a pendulum of length 18.5 m to swing from maximum displacement to equilibrium is approximately 2.17 seconds, which is a quarter of its period.
Explanation:
To find the time taken for a pendulum to swing from maximum displacement to the equilibrium position, we use the formula for the period of a simple pendulum, which is:
T = 2π√(L/g)
Where T is the period of oscillation, L is the length of the pendulum, and g is the acceleration due to gravity. However, the question asks for the time taken to swing from maximum displacement to equilibrium, which is only a quarter of the period. So, we divide the period by 4:
Time = T/4 = (1/2)π√(L/g)
Substitute the given values (L = 18.5 m and g = 9.81 m/s2):
Time = (1/2)π√(18.5/9.81)
After calculating, we find that the time taken is approximately 2.17 seconds.
A 129-kg horizontal platform is a uniform disk of radius 1.51 m and can rotate about the vertical axis through its center. A 67.5-kg person stands on the platform at a distance of 1.09 m from the center and a 25.3-kg dog sits on the platform near the person, 1.37 m from the center. Find the moment of inertia of this system, consisting of the platform and its population, with respect to the axis.
Answer:
I = 274.75 kg-m²
Explanation:
given,
mass of horizontal platform = 129 Kg
radius of the disk = 1.51 m
mass of the person(m_p) = 67.5 Kg
standing at distance(r_p) = 1.09 m
mass of dog(m_d) = 25.3 Kg
sitting near the person(r_d) = 1.37 m from center
moment of inertia = ?
Moment of inertia of disc = [tex]\dfrac{1}{2}MR^2[/tex]
Moment of inertia of the system
[tex]I = MOI\ of\ disk + m_p r_p^2 + m_d r_d^2[/tex]
[tex]I = \dfrac{1}{2}MR^2 + m_p r_p^2 + m_d r_d^2[/tex]
[tex]I = \dfrac{1}{2}\times 129 \times 1.51^2 + 67.5 \times 1.09^2 + 25.3\times 1.37^2[/tex]
I = 274.75 kg-m²
Calculate the pressure on the ground from an 80 kg woman leaning on the back of one of her shoes with a 1cm diameter heel, and calculate the pressure of a 5500 kg elephant with 20 cm diameter feet balancing on one foot.
Answer:
Pressure of woman will be [tex]99.87\times 10^5N/m^2[/tex]
Pressure of the elephant will be [tex]1716560.50N/m^2[/tex]
Explanation:
We have given that mass of the woman m = 80 kg
Acceleration due to gravity [tex]g=9.8m/sec^2[/tex]
Diameter of shoes = 1 cm =0.01 m
So radius [tex]r=\frac{d}{2}=\frac{0.01}{2}=0.005m[/tex]
So area [tex]A=\pi r^2=3.14\times 0.005^2=7.85\times 10^{-5}m^2[/tex]
We know that force is given F = mg
So [tex]F=80\times 9.8=784N[/tex]
Now we know that pressure is given by [tex]P=\frac{F}{A}=\frac{784}{7.85\times 10^{-5}}=99.87\times 10^5N/m^2[/tex]
Now mass of elephant m = 5500 kg
So force of elephant = 5500×9.8 = 53900 N
Diameter = 20 cm
So radius r = 10 cm
So area will be [tex]A=3.14\times 0.1^2=0.0314m^2[/tex]
So pressure will be [tex]P=\frac{53900}{0.0314}=1716560.50N/m^2[/tex]
Assuming that Albertine's mass is 60.0 kg , for what value of μk, the coefficient of kinetic friction between the chair and the waxed floor, does she just reach the glass without knocking it over? Use g = 9.80 m/s2 for the magnitude of the acceleration due to gravity.
Answer:
The value of coefficient of kinetic friction is 0.102.
Explanation:
Given that,
Mass of Albertine = 60.0 kg
Suppose Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8 m from her outstretched foot.
If the spring constant is 95.0 N/m.
We need to calculate the value of coefficient of kinetic friction
Using formula of work done
[tex]W =\dfrac{1}{2}kx^2[/tex]....(I)
The concept belongs to work energy and power
[tex]W=\mu g h[/tex]
Put the value into the formula
[tex]\dfrac{1}{2}kx^2=\mu g h[/tex]
Put the value into the formula
[tex]\dfrac{1}{2}\times95.0\times5.00^2=\mu\times60.0\times9.80\times19.8[/tex]
[tex]\mu=\dfrac{95.0\times5.00^2}{60.0\times9.80\times2\times19.8}[/tex]
[tex]\mu=0.102[/tex]
Hence, The value of coefficient of kinetic friction is 0.102.
The coefficient of kinetic friction between the chair and the waxed floor will be [tex]\mu_{k} =0.101[/tex]
What will be the coefficient of kinetic friction between the chair and the waxed floor /It is given that,
Mass of Albertine, m = 60 kg
It can be assumed, the spring constant of the spring, k = 95 N/m
Compression in the spring, x = 5 m
A glass sits 19.8 m from her outstretched foot, h = 19.8 m
When she just reaches the glass without knocking it over, a force of friction will also act on it. By conservation of energy, we can balance an equation
[tex]\dfrac{1}{2} kx^{2} =\mu_{k} mgh[/tex]
putting the values
[tex]\mu_{k} =\dfrac{kx^{2} }{2mgh}[/tex]
[tex]\mu_{k} =\dfrac{95\times 5^{2} }{2\times60\times9.8\times19.8}[/tex]
[tex]\mu_{k} =0.101[/tex]
Thus the coefficient of kinetic friction between the chair and the waxed floor will be [tex]\mu_{k} =0.101[/tex]
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Initially water is coming out of a hose at a velocity of 5 m/s. You place your thumb on the opening, reducing the area the water comes through to .001 $m^2$. You then find the velocity of the water to be 20 m/s. Calculate the radius of the hose.
Answer:
R = 0.001 m
Explanation:
Continuity equation
The continuity equation is nothing more than a particular case of the principle of conservation of mass. It is based on the flow rate (Q) of the fluid must remain constant throughout the entire pipeline.
Flow Equation
Q = v*A
where:
Q = Flow in (m³/s)
A is the surface of the cross sections of points 1 and 2 of the duct.
v is the flow velocity at points 1 and 2 of the pipe.
It can be concluded that since the Q must be kept constant throughout the entire duct, when the section (A) decreases, the speed (v) increases in the same proportion and vice versa.
Data
D₂= 0.001 m² : final hose diameter
v₁ = 5 m/s : initial speed of fluid
v₂ = 20 m/s : final speed of fluid
Area calculation
A = (π*D²)/4
A₁ = (π*D₁²)/4
A₂ = (π*D₂²)/4
Continuity equation
Q₁ = Q₂
v₁A₁ = v₂A₂
v₁(π*D₁²)/4 = v₂(π*D₂²)/4 : We divide by (π/4) both sides of the equation
v₁ (D₁)² = v₂(D₂)²
We replace data
5 *(D₁)² = 20*(0.001)²
(D₁)² = (20/5)*(0.001)²
(D₁)² = 4*10⁻⁶ m²
[tex]D_{1} = \sqrt{4*10^{-6} } ( m)[/tex]
D₁ = 2*10⁻³ m : diameter of the hose
Radius of the hose(R)
R= D₁/2
R= (2*10⁻³ m) / 2
R= (1*10⁻³ m) = 0.001 m
Pete slides a crate up a ramp with constant speed at an angle of 24.3 ◦ by exerting a 289 N force parallel to the ramp. How much work has been done against gravity when the crate is raised a vertical distance of 2.39 m? The coefficient of friction is 0.33. Answer in units of J.
Answer:
W=972.83 J
Explanation:
Given,
angle of inclination of ramp = θ = 24.3°
Force exerted on the block = F= 289 N
Crate is raised to height of = 2.39 m
coefficient of friction = 0.33
where g is the acceleration due to gravity = g = 9.8 m/s²
Work done = ?
let m be the mass of the crate
Gravity force along ramp = m g sinθ
Friction force = μm g cosθ
now writing all the force
F = m g sinθ + μm g cosθ
By putting the values
F = m g (sinθ + μcosθ)
289 = 9.8 x m ( sin 24.3°+ 0.33 cos 24.3°) ( take g =10 m/s²)
289 = 9.8 m(0.71)
m = 41.53 kg
Work done will be equal to
W= m g h
W= 41.53 x 9.8 x 2.39 J
W=972.83 J
Suppose that in the infinite square well problem, you set the potential energy at the bottom of the w ell to be equal to some constant V_0, rather than zero. Which of the following is the correct Schrodinger Equation inside the well? -h^2 partial differential^2 psi (x, t)/2 m partial differential x^2 = ih partial differential psi (x, t)/partial differential t -h^2 partial differential^2 psi (x, t)/2m partial differential x^2 + V_0 = ih partial differential psi (x, t)/partial differential t -h^2 partial differential^2 psi (x, t)/2 m partial differential x^2 - V_0 = ih partial differential psi (x, t)/partial differential t -h^2 partial differential^2 psi (x, t)/2m partial differential x^2 + V_0 psi (x, t) = ih partial differential psi (x, t)/partial differential t -*h^2 partial differential^2 psi (x, t)/2 m partial differential x^2 - V_0 psi (x, t) = ih partial differential psi (x, t)/partial differential t Which of the following are the correct energy levels for the infinite square well of width L with potential energy equal to V_0 at the bottom of the well? N^2 pi^2 h^2/2 mL^2 n^2pi^2h^2*/2mL^2 + V_0 n^2 pi^2 h^2/2mL^2 - V_0 V_) - n^2 pi^2 h^2/2 mL^2 n^2 pi^2 h^2/2 ml^2 V_0 V_0 n^2 pi^2 h^2/2 mL^2 None of the above - the potential energy has to be zero at the bottom of an infinite square well.
Answer:
Second equation, second energy are corrects
Explanation:
Schrödinger's equation is
-h’²/2m d²ψ/ dx² + V ψ = ih dψ / dt
Where h’= h/2π, h is the Planck constant, ψ the time-dependent wave function.
If we apply this equation to a well of infinite potential, there is only a solution within the well since, because the walls are infinite, electrons cannot pass to the other side,
In general we can place the origin of the regency system at any point, one of the most common to locate it at the bottom of the potential well so that V = 0, in this case it is requested that we place it lower so that V = V₀ , as this is an additive constant does not change the form of the solutions of the equation that is as follows
-h’²/2m d²ψ/dx² + Vo ψ = ih dψ / dt
Proposed Equations
First equation wrong missing the potential
Second equation correct
Third equation incorrect the power must be positive
Fifth. Incorrect is h ’Noel complex conjugate (* h’)
To find the potential well energy levels, we solve the independent equation of time
-h’² /2m d²φi /dx² + Vo φ = E φ
d²φ/ dx² = - 2m/h’² (E-Vo)φ
d²φ/dx² = k² φ
With immediate solution for being a second degree equation (harmonic oscillator), to be correct the solution must be zero in the well wall
φ = A sin k x
kL = √2m(E-Vo) /h’² = n pi
E- Vo = (h’²2 / 2mL²) n²
E = (h’² π² / 2 m L²) n² + Vo
Proposed Energies
First. wrong missing Vo
Second. correct
Third. Incorrect potential is positive, not a subtraction
Quarter. incorrect the potential is added energy is not a product
A thin, uniform, metal bar, 3.00 m long and weighing 90.0 N , is hanging vertically from the ceiling by a frictionless pivot. Suddenly it is struck 1.60 m below the ceiling by a small 3.00-kg ball, initially traveling horizontally at 10.0 m/s . The ball rebounds in the opposite direction with a speed of 5.00 m/s. Find the angular speed of the bar just after the collision? Why linear momentum not conserved?
Answer:
[tex]\omega_f=-2.6rad/s[/tex]
Since the bar cannot translate, linear momentum is not conserved
Explanation:
By conservation of the angular momentum:
Lo = Lf
[tex]I_B*\omega_o+I_b*(V_o/d)=I_B*\omega_f+I_b*(V_f/d)[/tex]
where
[tex]I_B =1/3*M_B*L^2[/tex]
[tex]I_b=m_b*d^2[/tex]
[tex]M_B=90/g=9kg[/tex]; [tex]m_b=3kg[/tex]; d=1.6m; L=3m; [tex]V_o=-10m/s[/tex]; [tex]V_f=5m/s[/tex]; [tex]\omega_o=0 rad/s[/tex]
Solving for [tex]\omega_f[/tex]:
[tex]\omega_f=m_b*d/I_B*(V_o-V_f)[/tex]
Replacing the values we get:
[tex]\omega_f=-2.6rad/s[/tex]
Since the bar can only rotate (it canno translate), only angular momentum is conserved.
A walrus transfers energy by conduction through its blubber at the rate of 150 W when immersed in −1.00ºC water. The walrus’s internal core temperature is 37.0ºC , and it has a surface area of 2.00 m^2. What is the average thickness of its blubber, which has the conductivity of fatty tissues without blood? (answer in cm)
Answer:
t=12.6 cm
Explanation:
Given that
Q= 150 W
T₁ = −1.00ºC
T₂ = 37.0ºC
A= 2 m²
Lets take thermal conductivity of blubber
K= 0.25 W/(m.K)
Lets take thickness = t
We know that heat transfer due to conduction given as
[tex]Q=KA\dfrac{(T_2-T_1)}{t}[/tex]
[tex]t=KA\dfrac{(T_2-T_1)}{Q}[/tex]
[tex]t=0.25\times 2\times \dfrac{(37 + 1)}{150}[/tex]
t=0.126 m
t= 12.6 cm
Therefore thickness of the blubber is 12.6 m.
1 Item 1 Item 1 1.25 points circular ceramic plate that can be modeled as a blackbody is being heated by an electrical heater. The plate is 30 cm in diameter and is situated in a surrounding ambient temperature of 15°C where the natural convection heat transfer coefficient is 12 W/m2·K. If the efficiency of the electrical heater to transfer heat to the plate is 80%, determine the electric power that the heater needs to keep the surface temperature of the plate at 180°C
Answer:
174.85 W
Explanation:
Area of plate = 3.14 x (15x 10⁻²)²
= 706.5 x 10⁻⁴ m²
heat being radiated by convection = 12 x 706.5 x 10⁻⁴ ( 180 - 15 )
= 139.88 W. This energy needs to be fed by heat source to maintain a constant temperature of 180 degree.
If power of electric source is P
P x .8 = 139.88
P = 139.88 / .8
= 174.85 W
Unpolarized light with intensity 300 W m2 is incident on three polarizers, P1, P2, and P3 numbered in the order light reaches the polarizers. The transmission axis of P1 and P2 make an angle of 45? . The transmission axis of P2 and P3 make an angle of 30? . How much light is transmitted through P3? Select One of the Following: (a) 60 W m2 (b) 100 W m2 (c) 20 W m2 (d) 10 W
Answer:
answer is A corresponding to 60 W
Explanation:
The polarization phenomenon is descriptor Malus's law
I = I₀ cos² θ
Where I and Io are the transmitted and incident intensities, respectively, θ is the angle between the direction of polarization of the light and the polarizer.
Unpolarized light strikes the first polarizer and half of it is transmitted, which has the polarization direction of the polarizer
I₁ = ½ I₀
I₁ = ½ 300
I₁ = 150 W
The transmission by the second polarizer (P2) is
I₂ = I₁ cos² θ 1
I₂ = 150 cos² 45
I₂ = 75 W
The transmission of the third polarizer (P3)
I₃ = I₂ cos² θ 2
I₃ = 75 cos² 30
I₃ = 56.25 W
This is the light transmitted by the 56 W system, the closest answer is A corresponding to 60 W
A circuit consists of a 9.0 mH inductor coil, a 230 Ω resistor, a 12.0 V ideal battery, and an open switch-all connected in series. The switch closes at time t = 0. What is the initial rate of change of the current - I just after the switch closes? dt l=0 A/s What is the steady-state value of the current If a long time after the switch is closed? If= At what time 185% does the current in the circuit reach 85% of its steady-state value? 185% = What is the rate of change of the current when the current in the circuit is equa dt li=1;/2 its steady-state value? dI - dt li=1;/2
Answer:
Explanation:
This is the case of L-R charging circuit , for which the formula is as follows
i = i₀ ( 1 - [tex]e^{\frac{-t}{\tau}[/tex] )
Differentiating the equation on both sides
di / dt = i₀ / τ x [tex]e^\frac{-t}{\tau}[/tex]
i is current at time t , i₀ is maximum current , τ is time constant which is equal to L / R where L is inductance and R is resistance of the circuit .
τ = L / R = 9 x 10⁻³ / 230
= 39 x 10⁻⁶ s
i₀ = 12 / 230 = 52.17 x 10⁻³
di / dt (at t is zero) = 52.17 x 10⁻³ / 39 x 10⁻⁶
= 1.33 x 10³ A / s
Time to reach 85 % of steady state or i₀
.85 = [tex]1 - e^\frac{t}{\tau}[/tex]
[tex]e^\frac{-t}{\tau}[/tex] = .15
t / τ = ln .15
t = 74 μs
An RL circuit with given parameters will have its steady-state current at 0.052A, with an initial rate of change of 1333.33 A/s. The current reaches 85% of its steady-state value after approximately 91µs. The rate of change when the current is half its steady-state value can be found using the I(t) function and its derivative.
Explanation:When you close a switch in an RL circuit, the current doesn't instantly reach its maximum value. Instead, it gradually increases until it reaches a steady-state value. This steady-state value, also known as the final current (If), can be calculated using Ohm's law, seeing that If = V(source voltage)/R(resistance). In your case, If = 12.0V/230Ω = 0.052A or 52mA.
The initial rate of change of current - dI/dt at t=0, can be calculated using the formula dI/dt = V(source voltage)/L(inductance). From your problem, dI/dt = 12.0V/9.0mH = 1333.33 A/s.
To calculate the time at which the current reaches 85% of its steady-state value, you use the formula for the time constant of an RL circuit, which is T = L/R. From the problem, T = 9.0mH/230Ω = 0.000039s, and the expression for time (t) when the current is at a certain percentage (p, in this case, 85%) of its steady-state value is t = -T*ln(1-p), which simplifies to t = -0.000039s*ln(1-0.85) = 0.000090999s or approximately 91µs.
The rate of change of the current when the current is at half its steady-state value will be smaller than the initial rate of change because the rate decreases as the current approaches its steady value. For the current at half its steady-state value, we calculate the time similar to the 85% case and find the I(t) at this time using I(t) = (V/R)(1-e^-Rt/L). Then, we differentiate it to find the rate of change dI/dt at this time.
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The total cross-sectional area of the load-bearing calcified portion of the two forearm bones (radius and ulna) is approximately 2.3 cm2. During a car crash, the forearm is slammed against the dashboard. The arm comes to rest from an initial speed of 80 km/h in 5.8 ms. If the arm has an effective mass of 3.0 kg, what is the compressional stress that the arm withstands during the crash?
To solve this problem, it is necessary to use the concepts related to the Force given in Newton's second law as well as the use of the kinematic equations of movement description. For this case I specifically use the acceleration as a function of speed and time.
Finally, we will describe the calculation of stress, as the Force produced on unit area.
By definition we know that the Force can be expressed as
F= ma
Where,
m= mass
a = Acceleration
The acceleration described as a function of speed is given by
[tex]a = \frac{\Delta v}{\Delta t}[/tex]
Where,
[tex]\Delta v =[/tex] Change in velocity
[tex]\Delta t =[/tex]Change in time
The expression to find the stress can be defined as
[tex]\sigma=\frac{F}{A}[/tex]
Where,
F = Force
A = Cross-sectional Area
Our values are given as
[tex]v= 80km/h\\t=5.8*10^3s\\m = 3kg \\A = 2.3*10^{-4}m^2[/tex]
Replacing at the values we have that the acceleration is
[tex]a = \frac{\Delta v}{\Delta t}[/tex]
[tex]a = \frac{80km/h(\frac{1h}{3600s})(\frac{1000m}{1km})}{5.8*10^3}[/tex]
[tex]a = 3831.41m/s^2[/tex]
Therefore the force expected is
[tex]F = ma\\F = 3*3831.41m/s^2 \\F = 11494.25N[/tex]
Finally the stress would be
[tex]\sigma = \frac{F}{A}[/tex]
[tex]\sigma = \frac{11494.25N}{2.3*10^{-4}}[/tex]
[tex]\sigma = 49.97*10^6 Pa = 49.97Mpa[/tex]
Therefore the compressional stress that the arm withstands during the crash is 49.97Mpa
Final answer:
The compressional stress is approximately 4.99787 × 10⁷ Pa
Explanation:
The question asks to find the compressional stress that the arm withstands during a crash, where the arm comes to a stop from an initial speed of 80 km/h in 5.8 ms, with an effective mass of 3.0 kg, and the total cross-sectional area of the load-bearing calcified portion of the forearm bones being approximately 2.3 cm².
First, convert the initial speed from km/h to m/s: 80 km/h = 22.22 m/s. To find the acceleration, use the formula a = ∆v / ∆t, where ∆v = -22.22 m/s (as it comes to rest) and ∆t = 5.8 ms or 0.0058 s. This gives an acceleration of approximately -3831.03 m/s².
The force experienced due to this acceleration can be calculated using Newton's second law, F = ma, with m = 3.0 kg and a = -3831.03 m/s², resulting in a force of approximately -11493.09 N. Finally, the compressional stress (σ) is found using the formula σ = F / A, where F = 11493.09 N and A = 2.3 cm² = 0.00023 m². This yields a compressional stress of approximately 4.99787 × 10⁷ Pa.
A 110-kg object and a 410-kg object are separated by 3.80 m.
(a) Find the magnitude of the net gravitational force exerted by these objects on a 41.0-kg object placed midway between them. N
(b) At what position (other than an infinitely remote one) can the 41.0-kg object be placed so as to experience a net force of zero from the other two objects?
Answer:
a) Fₙ = 2,273 10⁻⁷ N and b) x₃ = 1,297 m
Explanation:
This problem can be solved using the law of universal gravitation and Newton's second law for the equilibrium case. The Universal Gravitation Equation is
F = G m₁ m₂ / r₁₂²
a) we write Newton's second law
Σ F = F₁₃ - F₃₂
Body 1 has mass of m₁ = 110 kg and we will place our reference system, body 2 has a mass of m₂ = 410 kg and is in the position x₂ = 3.80 m
Body 3 has a mass of m₃ = 41.0 kg and is in the middle of the other two bodies
x₃ = (x₂-x₁) / 2
x₃ = 3.80 / 2 = 1.9 m
Fₙ = -G m₁ m₃ / x₃² + G m₃ m₂ / x₃²
Fₙ = G m₃ / x₃² (-m₁ + m₂)
Calculate
Fₙ = 6.67 10⁻¹¹ 41.0 / 1.9² (- 110 + 410)
Fₙ = 2,273 10⁻⁷ N
Directed to the right
b) find the point where the force is zero
The distance is
x₁₃ = x₃ - 0
x₃₂ = x₂ -x₃= 3.8 -x₃
We write the park equation net force be zero
0 = - F₁₃ + F₃₂
F₁₃ = F₃₂
G m₁ m₃ / x₁₃² = G m₃ m₂ / x₃₂²
m₁ / x₁₃² = m₂ / x₃₂²
Let's look for the relationship between distances, substituting
m₁ / x₃² = m₂ / (3.8 - x₃)²
(3.8 - x₃) = x₃ √ (m₂ / m₁)
x₃ + x₃ √ (m₂ / m₁) = 3.8
x₃ (1 + √ m₂ / m₁) = 3.8
x₃ = 3.8 / (1 + √ (m₂ / m₁))
x₃ = 3.8 / (1 + √ (410/110))
x₃ = 1,297 m
When body 3 is in this position the net force on it is zero
At t = 0, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 43.4 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?
Answer:30.08 ms
Explanation:
Given
time Constant [tex]\tau =43.4 ms =\frac{L}{R}[/tex]
Also rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in inductor's magnetic Field
Energy stored in Inductor is [tex]U_L=\frac{1}{2}Li^2[/tex]
rate of Energy storing [tex]\frac{dU_L}{dt}=\frac{1}{2}L\cdot 2i\times \frac{di}{dt}----1[/tex]
Rate of Energy dissipation from resistor i.e. Power is given by
[tex]\frac{dU_R}{dt}=i^2R-----2[/tex]
Equating 1 and 2
[tex]Li\cdot \frac{di}{dt}=i^2R[/tex]
[tex]L(\frac{di}{dt})=R(i)-----3[/tex]
i is given by [tex]i=\frac{V}{R}(1-e^{-\frac{t}{\tau }})[/tex]
[tex]\frac{di}{dt}=\frac{V}{L}e^{-\frac{t}{\tau }}[/tex]
substitute the value of [tex]\frac{di}{dt}[/tex] in 3
[tex]L(\frac{V}{L}e^{-\frac{t}{\tau }})=R\cdot \frac{V}{R}(1-e^{-\frac{t}{\tau }})[/tex]
[tex]e^{-\frac{t}{\tau }}=1-e^{-\frac{t}{\tau }}[/tex]
[tex]2e^{-\frac{t}{\tau }}=1[/tex]
[tex]e^{-\frac{t}{\tau }}=0.5[/tex]
[tex]e^{-\frac{t}{43.4\times 10^{-3}}}=0.5[/tex]
[tex]\frac{t}{43.4\times 10^{-3}}=0.693[/tex]
[tex]t=30.08 ms[/tex]