Find a glass jar with a screw-top metal lid. Close the lid snugly and put the jar into the refrigerator. Leave it there for about 10 minutes and then take the jar out and try to open the lid. (a) Did the lid become tighter or looser? Explain your observation.

Answer: 0.47 rad/sec

Explanation:

By definition, the angular velocity is the rate of change of the angle traveled with time, so we can state the following:

ω = ∆θ/ ∆t

Now, we are told that in 13.3 sec, the ball completes one revolution around the circle, which means that, by definition of angle, it has rotated 2 π rad (an arc of 2πr over the radius r), so we can find ω as follows:

ω = 2 π / 13.3 rad/sec = 0.47 rad/sec

Answer:

0.472rad/s

Explanation:

Angular velocity = 2πf where f = frequency and frequency is the number of revolution per second and 2π represent a cycle of revolution. The mass of the body was 5kg, the time taken to complete a cycle was 13.3 s.

Frequency = 1/period where period is the time it takes to complete a revolution.

F = 1/13.3 = 0.075hz

Angular velocity = 2* 3.142* 0.075 = 0.472rad/s

To solve this problem it is necessary to apply the concepts related to Dopler's Law. Dopler describes the change in frequency of a wave in relation to that of an observer who is in motion relative to the Source of the Wave.

It can be described as

[tex]f = \frac{c\pm v_r}{c\pm v_s}f_0[/tex]

c = Propagation speed of waves in the medium

[tex]v_r[/tex]= Speed of the receiver relative to the medium

[tex]v_s[/tex]= Speed of the source relative to the medium

[tex]f_0 =[/tex]Frequency emited by the source

The sign depends on whether the receiver or the source approach or move away from each other.

Our values are given by,

[tex]v_s = 32.2m/s \rightarrow[/tex] Velocity of car

[tex]v_r = 14.8 m/s \rightarrow[/tex] velocity of motor

[tex]c = 343m/s \rightarrow[/tex] Velocity of sound

[tex]f_0 = 523Hz \rightarrow[/tex]Frequency emited by the source

Replacing we have that

[tex]f = \frac{c + v_r}{c - v_s}f_0[/tex]

[tex]f = \frac{343 + 14.8}{343 - 32}(523)[/tex]

[tex]f = 601.7Hz[/tex]

Therefore the frequency that hear the motorcyclist is 601.7Hz

Final answer:

To determine the frequency heard by the motorcyclist, we apply the Doppler shift formula using the given speeds of the motorcycle and police car and the emitted frequency of the police car's siren. The calculated frequency is the observed frequency by the motorcyclist due to the relative motion of the two vehicles. The final answer is about 602.1 Hz

Explanation:

The subject of the question is the Doppler Effect, which is related to the change in frequency of sound waves due to the relative motion of the source and the observer. To calculate the frequency the motorcyclist hears, we use the following Doppler shift formula for sound:

f' = f(v + vo) / (v - vs)

Where:

f' is the observed frequency by the motorcyclist,

f is the emitted frequency by the police car (523 Hz),

v is the speed of sound in air, which can be assumed to be approximately 343 m/s at room temperature,

vo is the speed of the observer (motorcyclist) towards the source (14.8 m/s),

vs is the speed of the source (police car) towards the observer (32.2 m/s).

Plugging in the given values, the equation becomes:

f' = 523 Hz (343 m/s + 14.8 m/s) / (343 m/s - 32.2 m/s) = 602.089 Hz

After performing the calculations, the frequency heard by the motorcyclist can be determined. This is an application of the Doppler effect as studied in high school physics.

Answer:

0.04s

Explanation:

Specific heat of water c = 4.186J/g.C

The heat energy it would take to heat up 0.5 kg of water from 30C to 72 C is

[tex]E = mc\Delta T = 0.5 * 4.186 * (72 - 30) = 87.9 J[/tex]

If the power of the electric kettle is 2.2kW (or 2200W) and suppose the work efficiency is 100%, then the time it takes to transfer that power is

[tex] t = \frac{E}{P} = \frac{87.9}{2200} = 0.04 s[/tex]

Answer: Question 1: Efficiency is 0.6944

Question 2: speed of similar pump is 2067rpm

Explanation:

Question 1:

Flow rate of pump 1 (Q1) = 300gpm

Flow rate of pump 2 (Q2) = 400gpm

Head of pump (H)= 55ft

Speed of pump1 (v1)= 1500rpm

Speed of pump2(v2) = ?

Diameter of impeller in pump 1= 15.5in = 0.3937m

Diameter of impeller in pump 2= 15in = 0.381

B.H.P= 6.0

Assuming cold water, S.G = 1.0

eff= (H x Q x S.G)/ 3960 x B.H.P

= (55x 300x 1)/3960x 6

= 0.6944

Question 2:

Q = A x V. (1)

A1 x v1 = A2 x V2. (2)

Since A1 = A2 = A ( since they are geometrically similar

A = Q1/V1 = Q2/V2. (3)

V1(m/s) = r x 2π x N(rpm)/60

= (0.3937x 2 x π x 1500)/2x 60

= 30.925m/s

Using equation (3)

V2 = (400 x 30.925)/300

= 41.2335m/s

To rpm:

N(rpm) = (60 x V(m/s))/2 x π x r

= (60 x 41.2335)/ 2× π × 0.1905

= 2067rpm.

Answer:

Option C

Explanation:

The bob of the pendulum, when the pendulum is taken into the orbiting space station will be in free fall as there is no gravity in space and also the point to which it is attached.

The movement of the bob of the pendulum about its mean position is because of its weight due to which it oscillates and in the free fall, no force acts on it due to its weight since the body is not under the attraction force due to gravity thus it experiences weightlessness and does not oscillate.

On an orbiting space station with microgravity, a pendulum will not oscillate because the pendulum and its point of attachment are in free fall, absent of the normal gravitational forces.

If a pendulum is taken into an orbiting space station, its behavior will be altered due to the absence of an external gravitational force. This situation is best explained by option c. The pendulum will no longer oscillate because the pendulum and the point to which it is attached are in free fall, also known as microgravity. In an environment of microgravity, the forces that would ordinarily cause the pendulum bob to swing are absent, thus the pendulum will essentially float instead of oscillating back and forth.

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Answer:

[tex]8.2\times 10^{15}\ m[/tex]

Explanation:

[tex]\lambda[/tex] = Wavelength = 600 nm

d = Diameter of mirror = 42 m

D = Distance of object

x = Diameter of Jupiter = [tex]1.43\times 10^8\ m[/tex]

Angular resoulution is given by

[tex]\Delta\theta=1.22\frac{\lambda}{d}\\\Rightarrow \Delta\theta=1.22\frac{600\times 10^{-9}}{42}\\\Rightarrow \Delta\theta=1.74286\times 10^{-8}\ rad[/tex]

We also have the relation

[tex]\Delta\theta\approx=\frac{x}{D}\\\Rightarrow D\approx\frac{x}{\Delta\theta}\\\Rightarrow D\approx\frac{1.43\times 10^8}{1.74286\times 10^{-8}}\\\Rightarrow D\approx 8.2049\times 10^{15}\ m[/tex]

The most distant Jupiter-sized planet the telescope could resolve is [tex]8.2\times 10^{15}\ m[/tex]

Answer:

a) I = 13.77 A

b) 0 ° or to the East

Explanation:

Part a

The magnetic field by properties would be 0 at the radius on this case r =8.1 cm.Analyzing the situation the wirde would produce a magnetic field equals in magnitude to the magnetic field on Earth by with the inverse direction.

The formula for the magnetic field due to a wire with current is:

[tex] B = \frac{\mu_0 I}{2 \pi r} [/tex]

In order to have a value of 0 for the magnetic field at the radius then we need to have this balance

B (r=8.1) = B (Earth)

Replacing:

[tex] B = \frac{\mu_0 I}{2 \pi r)}= B_{Earth} [/tex]

Solving from I, from the last equation we got:

[tex] I = \frac{2 \pi r B_{earth}}{\mu_0} [/tex]

[tex] I=\frac{2 \pi 0.081 m (34 x 10^{-6} T)}{4 \pi x 10^{-7} Tm/A}[/tex] = 13.77 A

Part b

We can use the right hand rule for this case.

The magnetic field of the wire would point to the South, because the magnetic field of the earth given points to the North. Based on this the current need's to flow from West to East in order to create a magnetic field pointing to the south, because the current would be perpendicular to the magnetic field created.

Answer:(a)12.17 rad/s

Explanation:

Given

Moment of Inertia [tex]I=0.23 kg.m^2[/tex]

Torque [tex]T=2.8 N-m[/tex]

(a)Torque is given by Product of Moment of inertia and angular acceleration

[tex]T=I\cdot \alpha [/tex]

[tex]2.8=0.23\cdot \alpha [/tex]

[tex]\alpha =\frac{2.8}{0.23}=12.17 rad/s[/tex]

(b)RPM of blades [tex]N=205 rpm [/tex]

angular velocity [tex]\omega =\frac{2\pi N}{60}[/tex]

[tex]\omega =\frac{2\pi 205}{60}=21.47 rad/s[/tex]

Rotational Kinetic Energy [tex]=\frac{I\omega ^2}{2}[/tex]

[tex]=\frac{0.23\times (21.47)^2}{2}=53.01 J[/tex]

The angular acceleration of the blades is approximately 12.1739 rad/s^2. When the blades rotate at 205 rpm, the rotational kinetic energy is approximately 0.0948 J.

(a) To find the angular acceleration of the blades, we can use the formula:

torque = moment of inertia × angular acceleration

Plugging in the given values:

torque = 2.8 N · m

moment of inertia = 0.23 kg · m2

Rearranging the formula, we get:

angular acceleration = torque / moment of inertia

Substituting the values:

angular acceleration = 2.8 N · m / 0.23 kg · m2

Solving for angular acceleration gives us:

angular acceleration ≈ 12.1739 rad/s2

(b) To find the rotational kinetic energy, we can use the formula:

rotational kinetic energy = ½ × moment of inertia × (angular velocity)2

Plugging in the given values:

moment of inertia = 0.23 kg · m2

angular velocity = 205 rpm = 205 revolutions / 60 seconds = 3.4167 rev/s

Rearranging the formula, we get:

rotational kinetic energy = ½ × moment of inertia × (angular velocity)2

Substituting the values:

rotational kinetic energy = ½ × 0.23 kg · m2 × (3.4167 rev/s)2

Solving for rotational kinetic energy gives us:

rotational kinetic energy ≈ 0.0948 J

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Answer:

3.6 kHz

Explanation:

The pipes behave like a closed pipe . The end open is the end of the air canal outside the ear and the closed end is the eardrum.

The first harmonic will be as seen in the figure attached.

The length of the first harmonic will be λ/4.

λ/4=2.4 cm

λ=2.4 * 4=9.6 cm 0.096 m

Speed of Sound- 344 m/s(in air)

velocity(v) * Time Period(T) = Wavelength (λ)

Also, Time Period(T)= \frac{\textrm{1}}{\textrm{Frequency(f)}}

\frac{\textrm{Velocity}}{\textrm{Wavelength}}=\frac{\textrm{1}}{\textrm{Time Period}} =Frequency

Plugging in the values into the equation,

Frequency = [tex]\frac{344}{0.096}[/tex] Hz

                  = 3583.3 Hz≈3600 Hz= 3.6 kHz

Frequency= 3.6 kHz

Answer:

3.6 kHz

Explanation:

The auditory canal is a closed pipe because it has one closed end, the end terminated at the eardrum.

The length of the first harmonic of a closed pipe is given as;

L = λ/4 --------------------  (i)

where L = Length

and λ = wavelength

2.4 = λ/4

λ = 2.4 x 4 = 9.6 cm

Also, v = fλ  ------------------ (ii)

where v = speed of sound in air = 344 m/s

f = frequency of wave in Hertz

f = v/λ ------------------ (iii)

convert 9.6 cm to m = 0.096 m

substitute for λ and v in (iii)

[tex]f = \frac{344}{0.096} = 3583.33[/tex]

3583.33 Hz = 3600 Hz = 3.6 kHz

Answer:

0.0241875 m

Explanation:

[tex]m_1[/tex] = Mass of quarterback = 80 kg

[tex]m_2[/tex] = Mass of football = 0.43 kg

[tex]v_1[/tex] = Velocity of quarterback

[tex]v_2[/tex] = Velocity of football = 15 m/s

Time taken = 0.3 seconds

In this system as the linear momentum is conserved

[tex]m_1v_1+m_2v_2=0\\\Rightarrow v_1=-\frac{m_2v_2}{m_1}\\\Rightarrow v_1=-\frac{0.43\times 15}{80}\\\Rightarrow v_1=0.080625\ m/s[/tex]

Assuming this velocity is constant

[tex]Distance=Velocity\times Time\\\Rightarrow Distance=0.080625\times 0.3\\\Rightarrow Distance=0.0241875\ m[/tex]

The distance the quarterback will move in the horizontal direction is 0.0241875 m

Final answer:

The question is a physics problem regarding kinematics and requires calculating the horizontal distance an 80-kg quarterback moves while in the air after throwing a football. However, since the quarterback jumps vertically with zero initial horizontal speed and no horizontal force is acting after the throw, the resulting horizontal distance is zero.

Explanation:

The student is asking a question related to kinematics and the conservation of momentum in physics. Specifically, it concerns an 80-kg quarterback who jumps vertically and then throws a football horizontally. The core concept here is determining how far the quarterback will move horizontally during the time he is in the air, assuming a constant horizontal speed.

To calculate the distance (d) the quarterback moves horizontally, we can apply the equation for constant velocity motion: d = v * t. However, the information given does not directly provide the quarterback's horizontal speed after the throw; thus, we must assume that the horizontal speed of the quarterback is zero as the ball is thrown horizontally and no horizontal force on the quarterback has been mentioned.

Therefore, under the assumption that the quarterback's horizontal speed remains zero, the distance he moves horizontally d would also be zero, since he moves vertically up and down with no horizontal velocity component. If the problem had given a horizontal speed for the quarterback post-throw, we would use that given speed to calculate the distance using the equation.

Answer:

The final kinetic energy is 242 J.

Explanation:

Hi there!

According to the work-energy theorem, the total work done on the box is equal to its change in kinetic energy:

W = ΔKE

Where:

W = work done on the box.

ΔKE = change in kinetic energy (final KE - initial KE).

The only forces that do work in this case are the applied force and the friction force because the box moves only horizontally.

The equation of work is the following:

W = F · s

Where:

F = force.

s = traveled distance.

Then, the work done by the applied force is:

W = 122 N · 9.46 m = 1.15 × 10³ J

To calculate the work done by friction, we have to find the friction force:

Fr = N · μ

Where:

Fr = friction force.

N = normal force.

μ = coefficient of kinetic friction.

The box does not have a net vertical acceleration. It means that the sum of the vertical forces acting on the box is zero:

∑Fy = 0

In this case, the only vertical forces are the weight of the box and the normal force. Then:

Weight + N = 0

N = - Weight

The weight of the box is calculated as follows:

Weight = m · g

Where:

m = mass of the box.

g = acceleration due to gravity.

Then:

-Weight = N = 28 kg · 9.8 m/s² = 274.4 N

Now, we can calculate the friction force:

Fr = N · μ

Fr = 274.4 N · 0.35 = 96 N

The work done by the friction force will be:

W = Fr · s

W = 96 N · 9.46 m = 908 J

Since the work done by friction opposes to the sense of movement, the work is negative.

Now, we can calculate the total work done on the box:

W total = W applied forece + W friction force

W = 1.15 × 10³ J - 908 J = 242 J

Applying the work-energy theorem:

W = final KE - initial KE

Since the box is initially at rest the initial kinetic energy is zero. Then:

W = final KE - 0

W = final KE

Final KE = 242 J

The final kinetic energy is 242 J.

Answer:

a) the velocity is v=1.385 m/s

b) the ball has its maximum speed at 4.68 cm away from its compressed position

c)  the maximum speed is 1.78 m/s

Explanation:

if we do an energy balance over the ball, the potencial energy given by the compressed spring is converted into kinetic energy and loss of energy due to friction, therefore

we can formulate this considering that the work of the friction force is equal to to the energy loss of the ball

W fr = - ΔE = - ΔU - ΔK = Ui - Uf + Kf - Ki

therefore

Ui + Ki = Uf + Kf + W fr  

where U represents potencial energy of the compressed spring , K is the kinetic energy W fr is the work done by the friction force. i represents inicial state, and f final state.

since

U= 1/2 k x² , K= 1/2 m v²  , W fr = F*L

X= compression length , L= horizontal distance covered

therefore

Ui + Ki = Uf + Kf + W fr

1/2 k xi² + 1/2 m vi² = 1/2 k x² + 1/2 m vf² + F*L

a) choosing our inicial state as the compressed state , the initial kinetic energy is Ki=0 and in the final state the ball is no longer pushed by the spring thus Uf=0

1/2 k X² + 0 = 0 + 1/2 m v² + F*L

1/2 m v² = 1/2 k X² - F*L

v = √[(k/m)x² -(2F/m)*L] = √[(8.07N/m/5.35*10^-3 Kg)*(-0.0508m)² -(2*0.033N/5.35*10^-3 Kg)*(0.16 m)] = 1.385 m/s

b) in any point x , and since L= d-(X-x) , d = distance where is no pushed by the spring.

1/2 k X² + 0 = 1/2 k x² + 1/2 m v² + F*[d-(X+x)]

1/2 m v² =1/2 k X²-1/2 k x² - F*[d-(X-x)] = (1/2 k X²+ F*X) - 1/2k x² - F*x + F*d

taking the derivative

dKf/dx = -kx - F = 0 → x = -F/k = -0.033N/8.07 N/m = -4.089*10^-3 m = -0.4cm

at x m = -0.4 cm the velocity is maximum

therefore is 5.08 cm-0.4 cm=4.68 cm away from the compressed position

c) the maximum speed is

1/2 m v max² = (1/2 k X²+ F*X) - 1/2k x m² - F*(x m) + 0

v =√[ (k/m) (X²-xm²) + (2F/m)(X-xm) ] = √[(8.07N/m/5.35*10^-3 Kg)*[(-0.0508m)² - (-0.004m)²] + (2*0.033N/5.35*10^-3 Kg)*(-0.0508m-(-0.004m)] = 1.78 m/s

Final answer:

When the glass flask and mercury are warmed together, the mercury expands and overflows the flask. To calculate the initial volume of the mercury, use the equation: Volume of mercury = Volume of flask + Volume overflowed.

Explanation:

When the glass flask and mercury are warmed together, both substances expand due to the increase in temperature. As a result, some of the mercury overflows the flask. To calculate the initial volume of the mercury, we can use the equation:

Volume of mercury = Volume of flask + Volume overflowed

So, the initial volume of mercury is 1000 cm³ + 8.25 cm³ = 1008.25 cm³.

Answer:

each door has a width of 2.666 meters

Explanation:

from Newton's second law applied to rotational motion:

ζ = I α

where ζ= torque , I = moment of inertia , α = angular acceleration

the moment of inertia for a flat plate around its central axis is

I = 1/12 m a² , where m= mass, a= total width = 2L

therefore the moment of inercia for a flat plate with length 2L ( 2 doors, one in each side of the central axis) is

I1 = 1/12 m (2L)² = 1/3 m L²

if we have 4 doors , that is 2 flat plates with length of 2L perpendicular to each other:

I = Ix + Iy = 2*I1 = 2/3 m L²

thus

ζ = I α

4* F * L = 2/3 * (4*m) L² * α

L = 3/2* F/ ( m*α) = 3/2* 56 N / ( 75 Kg * 0.420 rad/seg²) = 2.666 m

Answer:

Specific Gravity = m/[m(s)-m(os)]

Explanation:

Specific gravity, also called relative density, is the ratio of the density of a substance to the density of a reference substance. By this definition we need to find out the ratio of density of the object of mass m to the density of the surrounding liquid.

m = mass of the object

Weight in air

W (air) = mg, where g is the gravitational acceleration

Weight with submerged with only one mass

m(s)g + Fb = mg + m(b)g, consider this to be equation 1

where Fb is the buoyancy force

Weight with submerged with both masses

m(os)g + Fb’ = mg + m(b)g, consider this to be equation 2

equation 1 – equation 2 would give us

m(s)g – m(os)g = Fb’ – Fb

where Fb = D x V x g, where D is the density of the liquid the object is submerged in, g is the force of gravity and V is the submerged volume of the object

m(s)g – m(os)g = D(l) x V x g

m(s) – m(os) = D(l) x V

we know that Mass = Density x V, which in our case would be, D(b) x V, which also means

V = Mass/D(b), where D(b) is the density of the mass

Substituting V into the above equation we get

m(s) – m(os) = [D(l) x m)/ D(b)]

Rearranging to get the ratio of density of object to the density of liquid

D(b)/D(l) = m/[m(s)-m(os)], where D(b)/D(l) denotes the specific gravity

Answer:

4.6939 rad/s

Explanation:

You have to use the conservation of angular momentum for both objects as an object is spinning and a collision happens. To properly solve, you have to know that the putty and the vinyl have the same angular speed after the putty lands on it and that the putty acts as a point mass so the formula of it's rotational inertia is mr^2.

Answer:

Choice A. The bouncy "superball" will exert approximately twice as much force than the blob of clay when tossed out at the same speed.

Explanation:

The momentum of an object is equal to the product of its mass and its velocity. The momentum of the system is conserved even if the collision is inelastic.

When the bouncy ball or the blob of clay hits the door, their momentum changes. Since momentum is conserved, the door would move in the opposite direction such that the total momentum stays at zero.

Let the mass of each object be [tex]m[/tex]. Suppose that both objects hit the door at a speed of [tex]v[/tex] in the same direction.

By Newton's Second Law, the net force on the door is proportion to the rate of change in its momentum. Assume that the two objects are in contact with the door for the same amount of time, the bouncy ball would exert about twice as much force on the door as the clay would. Hence it is "more effective" for closing the door.

The clay is more effective than the superball for shutting the door because upon impact the clay sticks to the door and transfers all of its momentum to it, while the superball bounces back and doesn't transfer all of its momentum.

The most effective object to throw at your door to shut it would be c. The blob of clay. This is because of the principle of conservation of momentum. When the clay hits the door, it sticks to it, thus transferring all of its momentum to the door. This momentum is what causes the door to move and potentially close. On the other hand, when the superball hits the door, it bounces back, meaning not all of its momentum is transferred to the door. Hence, the clay, by virtue of its ability to stick and transfer its full momentum, would be more effective.

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The tangential acceleration of the pedal is 0.259 m/s^2. The length of chain passing over the top of the sprocket depends on the number of teeth on the sprocket, which is not provided in the question.

A) The tangential acceleration of the pedal can be determined using the equation:

at = r × α

where at is the tangential acceleration, r is the radius of the pedal (half the length of the crank arm), and α is the angular acceleration. To find the angular acceleration, we can use the equation:

α = Δω / Δt

where Δω is the change in angular velocity and Δt is the change in time. Plugging in the given values:

α = (95 rpm - 62 rpm) * 2π / 60 s = (33 * 2π) / 60 s ≈ 3.459 rad/s2

Substituting this value and the radius into the first equation, we get:

at = 0.075 m * 3.459 rad/s2 = 0.259 m/s2

Therefore, the tangential acceleration of the pedal is 0.259 m/s2.

B) The length of chain passing over the top of the sprocket during the given time interval can be calculated using the formula:

d = 2π * r * (ωf - ωi) * t / N

where d is the length of chain, r is the radius of the sprocket, ωi and ωf are the initial and final angular velocities respectively, t is the time interval, and N is the number of teeth on the sprocket. Plugging in the given values:

d = 2π * (21 cm / 2) * [(95 rpm - 62 rpm) * 2π / 60 s] * 12 s / N

The length of chain depends on the number of teeth on the sprocket. Since the number of teeth is not provided in the question, we cannot calculate the exact length of chain. However, we have all the necessary equation components to calculate it once the number of teeth is known.

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Answer:

217.43298 m/s

Explanation:

[tex]m_1[/tex] = Mass of bullet = 19 g

[tex]m_2[/tex] = Mass of bob = 1.3 kg

L = Length of pendulum = 2.3 m

[tex]\theta[/tex] = Angle of deflection = 60°

u = Velocity of bullet

Combined velocity of bullet and bob is given by

[tex]v^2-u^2=2as\\\Rightarrow v=\sqrt{2aL(1-cos\theta)+u^2}\\\Rightarrow v=\sqrt{2\times 9.81\times (1-cos60)+0^2}\\\Rightarrow v=3.13209\ m/s[/tex]

As the momentum is conserved

[tex]m_1u=(m_1+m_2)v\\\Rightarrow u=\frac{(m_1+m_2)v}{m_1}\\\Rightarrow v=\frac{(0.019+1.3)\times 3.13209}{0.019}\\\Rightarrow v=217.43298\ m/s[/tex]

The speed of the bullet is 217.43298 m/s

The problem involves conservation of momentum and energy principles. Initially, bullet's momentum equals the final momentum of the system. The bullet's speed can be found by solving these equations, using the provided values.

This problem can be solved using principles from both conservation of momentum and conservation of energy. To find the speed of the bullet, we need to consider two scenarios: the before and after the bullet is fired into the bob. Initial momentum is the mass of the bullet multiplied by its velocity and final momentum is the combined mass of the bullet and bob at their highest point. Assuming there's no external force acting, we can have:

m_bullet * v_bullet = (m_bullet + m_bob) * v_final.

The final velocity here is the vertical component of the velocity when the pendulum reach its highest point. This can be calculated by:

v_final = sqrt(2*gravity*height).

The height can be calculated using trigonometry:

height = length - length * cos(60).

Filling all the given values into the equations will give the speed of the bullet.

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Answer:

The deformation in the pole due to force is 0.70 mm.

Explanation:

Given that,

Height = 20.0 m

Diameter = 4.00 cm

Force = 1100 N

We need to calculate the  area

Using formula of area

[tex]A=\pi\times r^2[/tex]

[tex]A=\pi\times(2.00\times10^{-2})^2[/tex]

[tex]A=0.00125\ m^2[/tex]

[tex]A=1.25\times10^{-3}\ m^2[/tex]

We need to calculate the deformation

Using formula of deformation

[tex]\Delta x=\dfrac{1}{s}(\dfrac{F}{A}\times L)[/tex]

Where, s = shear modulus

F = force

l = length

A = area

Put the value into the formula

[tex]\Delta x=\dfrac{1}{2.5\times10^{10}}\times(\dfrac{1100}{1.25\times10^{-3}}\times 20.0)[/tex]

[tex]\Delta x=0.000704\ m[/tex]

[tex]\Delta x=7.04\times10^{-4}\ m[/tex]

[tex]\DElta x=0.70\ mm[/tex]

Hence, The deformation in the pole due to force is 0.70 mm.

Answer:

a) C.M [tex]=(\bar x, \bar y)=(0.767,0.7)m[/tex]

b) [tex](x_4,y_4)=(-1.917,-1.75)m[/tex]

Explanation:

The center of mass "represent the unique point in an object or system which can be used to describe the system's response to external forces and torques"

The center of mass on a two dimensional plane is defined with the following formulas:

[tex]\bar x =\frac{\sum_{i=1}^N m_i x_i}{M}[/tex]

[tex]\bar y =\frac{\sum_{i=1}^N m_i y_i}{M}[/tex]

Where M represent the sum of all the masses on the system.

And the center of mass C.M [tex]=(\bar x, \bar y)[/tex]

Part a

[tex]m_1= 3 kg, m_2=5kg,m_3=7kg[/tex] represent the masses.

[tex](x_1,y_1)=(0,0),(x_2,y_2)=(2.3,0),(x_3,y_3)=(0,1.5)[/tex] represent the coordinates for the masses with the units on meters.

So we have everything in order to find the center of mass, if we begin with the x coordinate we have:

[tex]\bar x =\frac{(3kg*0m)+(5kg*2.3m)+(7kg*0m)}{3kg+5kg+7kg}=0.767m[/tex]

[tex]\bar y =\frac{(3kg*0m)+(5kg*0m)+(7kg*1.5m)}{3kg+5kg+7kg}=0.7m[/tex]

C.M [tex]=(\bar x, \bar y)=(0.767,0.7)m[/tex]

Part b

For this case we have an additional mass [tex]m_4=6kg[/tex] and we know that the resulting new center of mass it at the origin C.M [tex]=(\bar x, \bar y)=(0,0)m[/tex] and we want to find the location for this new particle. Let the coordinates for this new particle given by (a,b)

[tex]\bar x =\frac{(3kg*0m)+(5kg*2.3m)+(7kg*0m)+(6kg*a)}{3kg+5kg+7kg+6kg}=0m[/tex]

If we solve for a we got:

[tex](3kg*0m)+(5kg*2.3m)+(7kg*0m)+(6kg*a)=0[/tex]

[tex]a=-\frac{(5kg*2.3m)}{6kg}=-1.917m[/tex]

[tex]\bar y =\frac{(3kg*0m)+(5kg*0m)+(7kg*1.5m)+(6kg*b)}{3kg+5kg+7kg+6kg}=0m[/tex]

[tex](3kg*0m)+(5kg*0m)+(7kg*1.5m)+(6kg*b)=0[/tex]

And solving for b we got:

[tex]b=-\frac{(7kg*1.5m)}{6kg}=-1.75m[/tex]

So the coordinates for this new particle are:

[tex](x_4,y_4)=(-1.917,-1.75)m[/tex]

Answer:

P = 1.89 10⁸ N / m²

Explanation:

To solve this problem we can use the definition of bulk modules

       B = - P / (ΔV/V)

The negative sign is entered for the volume module to be positive, P is the pressure and ΔV/V is the volume change fraction

In this case the volume change is 9% this is

      ΔV / V 100 = 9%

      ΔV / V = ​​0.09

      P = B ΔV / V

The bulk modulus value is that of water since it is in a liquid state and then freezes

       B = 0.21 101¹⁰ N / m²

let's calculate

       P = 0.21 10¹⁰ 0.09

      P = 1.89 10⁸ N / m²

To solve this problem it is necessary to use the concepts related to Snell's law.

Snell's law establishes that reflection is subject to

[tex]n_1sin\theta_1 = n_2sin\theta_2[/tex]

Where,

[tex]\theta =[/tex] Angle between the normal surface at the point of contact

n = Indices of refraction for corresponding media

The total internal reflection would then be given by

[tex]n_1 sin\theta_1 = n_2sin\theta_2[/tex]

[tex](1.54) sin\theta_1 = (1.33)sin(90)[/tex]

[tex]sin\theta_1 = \frac{1.33}{1.54}[/tex]

[tex]\theta = sin^{-1}(\frac{1.33}{1.54})[/tex]

[tex]\theta = 59.72\°[/tex]

Therefore the [tex]\alpha_{max}[/tex] would be equal to

[tex]\alpha = 90\°-\theta[/tex]

[tex]\alpha = 90-59.72[/tex]

[tex]\alpha = 30.27\°[/tex]

Therefore the largest value of the angle α is 30.27°

The largest value the angle α can have without any light being refracted out of the prism at face AC is 30.27°.

From the information given, the total internal reflection would be:

1.54sinb = (1.33) sin90°

sin b = (1.33 sin90° / 1.54)

b = 59.72°

Therefore, the value of the angle will be:

= 90° - 59.72°

= 30.27°

In conclusion, the correct option is 30.27°.

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Answer:

Explanation:

Tension T in the rope will create torque in solid cylinder ( axle ). If α be angular acceleration

T R = 1/2 M R²α ( M is mass and R is radius of cylinder )

= 1/2 M R² x a / R ( a is linear acceleration )

T = Ma / 2

For downward motion of the bucket

mg - T = m a ( m is mass and a is linear acceleration of bucket downwards )

mg - Ma / 2  = ma

a = mg / ( M /2 + m )

Substituting the values

a = 14.7 x 9.8 / ( 5.8+ 14.7 )

= 7 m / s²

A )

T = Ma / 2

= 5.8 x 7

= 40.6 N

B ) v² = u² + 2 a h

= 2 x 7 x 10.3

v = 12 m /s

C )

v = u + a t

12 = 0 + 7 t

t = 1.7 s

Answer:

[tex]v=346.05\ m.s^{-1}[/tex]

Explanation:

Given:

initial temperature of the lead bullet, [tex]T_i=43^{\circ}C[/tex]

latent heat of fusion of lead, [tex]L_f=2.32\times 10^4\ J.kg^{-1}[/tex]

melting point of lead, [tex]T_m=327.3^{\circ}C[/tex]

We have:

specific heat capacity of lead, [tex]c=129\ J.kg^{-1}.K^{-1}[/tex]

According to question the whole kinetic energy gets converted into heat which establishes the relation:

[tex]\rm KE=(heat\ of\ rising\ the\ temperature\ from\ 43\ to\ 327.3\ degree\ C)+(heat\ of\ melting)[/tex]

[tex]\frac{1}{2} m.v^2=m.c.\Delta T+m.L_f[/tex]

[tex]\frac{1}{2} m.v^2=m(c.\Delta T+L_f)[/tex]

[tex]\frac{v^2}{2} =129\times(327.3-43)+23200[/tex]

[tex]v=346.05\ m.s^{-1}[/tex]

The minimum speed of a lead bullet that will completely melt upon stopping suddenly can be found by calculating the kinetic energy that is converted into heat via friction and comparing it with the heat required to elevate the bullet's temperature to its melting point. The formula for calculating the minimum speed of the bullet is the square root of twice the product of the latent heat of fusion and the change in temperature (√(2Lf∆T)).

The question asked pertains to the concept of kinetic energy transformations and heat generation due to friction in Physics. The kinetic energy of the bullet is transformed into thermal energy due to the sudden change in speed, causing the lead bullet to heat up and potentially melt if it is moving fast enough.

First, we calculate the change in temperature which is the difference between the melting point of lead and the initial temperature, ∆T = 327.3°C - 43.0°C = 284.3°C. Then, we utilize the formula for heat transfer Q = mLf, where m is the mass of the bullet, and Lf is the latent heat of fusion of lead. We rearrange this to find the mass of the bullet, m = Q/Lf.

Next, we use the principle of conservation of energy. All of the kinetic energy of the bullet (1/2mv²) is converted into heat (Q), leading to the equation 1/2m v² = Q. Solving for v (the bullet's speed) we get that v = √(2Q/m). Combining equations yields v = √(2Lf∆T).

This v is the minimum speed at which the bullet will completely melt upon stopping suddenly, assuming that all of its kinetic energy is converted into heat via friction and is equal to the energy required to raise the bullet's temperature to its melting point.

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Answer:

29274.93096 m/s

[tex]2.73966\times 10^{33}\ J[/tex]

[tex]-5.39323\times 10^{33}\ J[/tex]

[tex]2.56249\times 10^{33}\ J[/tex]

[tex]-5.21594\times 10^{33}[/tex]

Explanation:

[tex]r_p[/tex] = Distance at perihelion = [tex]1.471\times 10^{11}\ m[/tex]

[tex]r_a[/tex] = Distance at aphelion = [tex]1.521\times 10^{11}\ m[/tex]

[tex]v_p[/tex] = Velocity at perihelion = [tex]3.027\times 10^{4}\ m/s[/tex]

[tex]v_a[/tex] = Velocity at aphelion

m = Mass of the Earth =  5.98 × 10²⁴ kg

M = Mass of Sun = [tex]1.9889\times 10^{30}\ kg[/tex]

Here, the angular momentum is conserved

[tex]L_p=L_a\\\Rightarrow r_pv_p=r_av_a\\\Rightarrow v_a=\frac{r_pv_p}{r_a}\\\Rightarrow v_a=\frac{1.471\times 10^{11}\times 3.027\times 10^{4}}{1.521\times 10^{11}}\\\Rightarrow v_a=29274.93096\ m/s[/tex]

Earth's orbital speed at aphelion is 29274.93096 m/s

Kinetic energy is given by

[tex]K=\frac{1}{2}mv_p^2\\\Rightarrow K=\frac{1}{2}\times 5.98\times 10^{24}(3.027\times 10^{4})^2\\\Rightarrow K=2.73966\times 10^{33}\ J[/tex]

Kinetic energy at perihelion is [tex]2.73966\times 10^{33}\ J[/tex]

Potential energy is given by

[tex]P=-\frac{GMm}{r_p}\\\Rightarrow P=-\frac{6.67\times 10^{-11}\times 1.989\times 10^{30}\times 5.98\times 10^{24}}{1.471\times  10^{11}}\\\Rightarrow P=-5.39323\times 10^{33}[/tex]

Potential energy at perihelion is [tex]-5.39323\times 10^{33}\ J[/tex]

[tex]K=\frac{1}{2}mv_a^2\\\Rightarrow K=\frac{1}{2}\times 5.98\times 10^{24}(29274.93096)^2\\\Rightarrow K=2.56249\times 10^{33}\ J[/tex]

Kinetic energy at aphelion is [tex]2.56249\times 10^{33}\ J[/tex]

Potential energy is given by

[tex]P=-\frac{GMm}{r_a}\\\Rightarrow P=-\frac{6.67\times 10^{-11}\times 1.989\times 10^{30}\times 5.98\times 10^{24}}{1.521\times 10^{11}}\\\Rightarrow P=-5.21594\times 10^{33}[/tex]

Potential energy at aphelion is [tex]-5.21594\times 10^{33}\ J[/tex]

Answer:

:) *_* :3 ^-^ {.}{.}

Explanation:

To solve this problem we can use the concepts related to the change of flow of a fluid within a tube, which is without a rubuleous movement and therefore has a laminar fluid.

It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law.

The mathematical equation that expresses this concept is

[tex]\dot{Q} = \frac{\pi r^4 (P_2-P_1)}{8\eta L}[/tex]

Where

P = Pressure at each point

r = Radius

[tex]\eta =[/tex] Viscosity

l = Length

Of all these variables we have so much that the change in pressure and viscosity remains constant so the ratio between the two flows would be

[tex]\frac{\dot{Q_A}}{\dot{Q_B}} = \frac{r_A^4}{r_B^4}\frac{L_B}{L_A}[/tex]

From the problem two terms are given

[tex]R_A = \frac{R_B}{2}[/tex]

[tex]L_A = 2L_B[/tex]

Replacing we have to

[tex]\frac{\dot{Q_A}}{\dot{Q_B}} = \frac{r_A^4}{r_B^4}\frac{L_B}{L_A}[/tex]

[tex]\frac{\dot{Q_A}}{\dot{Q_B}} = \frac{r_B^4}{16*r_B^4}\frac{L_B}{2*L_B}[/tex]

[tex]\frac{\dot{Q_A}}{\dot{Q_B}} = \frac{1}{32}[/tex]

Therefore the ratio of the flow rate through capillary tubes A and B is 1/32

Answer:

[tex]a_{cm} = 9.64m/s^2[/tex]

[tex]Ff=6.42N[/tex]

Explanation:

The sum of torque on the sphere is:

[tex]m*g*sin\theta*R=I*\alpha[/tex]

[tex]m*g*sin\theta*R=2/3*m*R^2*\alpha[/tex]

[tex]m*g*sin\theta*R=2/3*m*R*a_{cm}[/tex]

Solving for a:

[tex]a_{cm}=9.64m/s^2[/tex]

Now, the sum of forces will be:

[tex]m*g*sin\theta-Ff=m*a_{cm}[/tex]

Solving for Ff:

[tex]Ff=m*g*sin\theta-m*a_{cm}[/tex]

Ff=-6.42N    The negative sing tells us that it actually points downwards.

Answer:

a) a = 3.783 m/s^2

b)  F_f = 5.045 N

Explanation:

Given:

- Mass of shell m = 2.0 kg

- Angle of slope Q = 40 degrees

- Moment of inertia of shell I = 2/3 *m*R^2

Find:

a) Find the magnitude of the acceleration a_cm of the center of mass of the spherical shell.

b) Find the magnitude of the frictional force acting on the spherical shell.

Solution:

- Draw a Free body diagram for the shell. We see that the gravitational force F_g acting parallel to the plane of the inclined surface makes the sphere to roll down. The frictional force  F_f between the inclined surface and the sphere gives the necessary torque for the sphere to roll down with out slipping. Under this conditions a sphere will roll down without slipping with some acceleration and the acceleration can be calculated from the equation of motion of the sphere:

                                      m*g*sin(Q) - F_f = m*a

- Where, The frictional force produces the torque and due to this torque the sphere gets an angular acceleration.

- Then we can write the equation for the rotational motion as:

                                      F_f*R = I*α

                                      F_f = I*α / R

- Using moment mass inertia of the shell we have:

                                      F_f = (2/3)*m*R^2*α/R

- Where the angular acceleration α is related to linear acceleration a with:

                                      α = a / R

- combing the two equations we will have friction force F_f as:

                                      F_f = (2/3)*m*R^2*a/R^2

                                      F_f = (2/3)*m*a

- Now evaluate the equation of motion:

                                      m*g*sin(Q) - (2/3)*m*a= m*a

- Simplify:

                                        (5/3)*a = g*sin(Q)

                                        a = (3/5)*g*sin(Q)

- Plug the values in:       a = (3/5)*9.81*sin(40)

                                        a = 3.783 m/s^2

- Now compute the Frictional force F_f from the expression derived above:

                                        F_f = (2/3)*m*a

- Plug values in:              F_f = (2/3)*2*3.783

                                        F_f = 5.045 N

Answer:

Explanation:

initial velocity, u = 28 m/s

Angle of projection, θ = 30°

The acceleration in horizontal direction is zero, so the horizontal component of velocity is constant.

Horizontal component of velocity, u cos θ = 28 x Cos 30 = 24.25 m/s

At t = 2 sec, the horizontal component of velocity = 24.25 m/s

At t = 3 sec, the horizontal component of velocity = 24.25 m/s

In this exercise we have to use the knowledge about oblique launch to calculate the components of velocity for each case, so we have that:

For all times we will find a velocity equal to 24.25 m/s

organizing the information given in the statement we have that:

Knowing that the component can be written as:

[tex]u cos \theta = 28 * Cos 30 = 24.25 m/s[/tex]

So as the formula does not depend on time we have that for any value it will have a constant velocity.

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Answer:

Explanation:

Let equal mass of Ne and Kr be m gm

no of moles of Ne and Kr will be m / 20  and m / 84 ( atomic weight of Ne and Kr is 20 and 84 )

Let the pressure and volume of both the gases be P and V respectively .

The temperature of Ne be T₁ and temperature of Kr be T₂.

For Ne

PV = (m / 20) x R T₁

For Kr

PV = (m / 84) x R T₂

T₁ / T₂ = 84 / 20

We know that

average KE of an atom of mono atomic gas = 3 / 2 x k T

k is boltzmann constant and T is temperature .

KEKr/KENe = T₂ / T₁

= 20 / 84

The ratio of the average kinetic energy of an atom of krypton to that of neon is 1.

The average kinetic energy of an atom is directly proportional to the temperature. In this case, the temperatures of the krypton (Kr) and neon (Ne) gases are different. Since the temperatures are different, the ratio of their average kinetic energies will be equal to the ratio of their temperatures. Therefore, the ratio KEKr/KENe is equal to Tkrypton/Tneon.

Using the given information, we can find the ratio of the temperatures: Tkrypton/Tneon = 300K/300K = 1. Therefore, the ratio of the average kinetic energy of an atom of krypton to that of neon is 1.

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