AP Physics C Mechanics Translational Kinetic Energy FRQ

Translational Kinetic Energy AP  Physics C Mechanics FRQ – Exam Style Questions etc.

Translational Kinetic Energy AP  Physics C Mechanics FRQ

Unit 3: Work, Energy, and Power

Weightage : 15-25%

AP Physics C Mechanics Exam Style Questions – All Topics

Question 

Block A and Block B of masses m and 3m, respectively, are arranged in a setup consisting of an ideal spring
with spring constant k and a horizontal surface. Friction between the surface and the blocks is negligible except in
a region of length D, where the coefficient of kinetic friction between Block A and the surface is m. Block B is
attached to a string of length and negligible mass, as shown in Figure 1. Block A is held against the spring,
compressing the spring a distance xc.

At time=0, Block A is located at position x=xο and is released from rest. After the block is released, the following occurs.
• At time $t = t_{1}$, Block A is at$ x =x_{1}$ after traveling a distance xc. Block A moves with speed v, and the spring is at its equilibrium position.
• At time $t = t_{1}$, the left side of Block A is at$ x = x_{1}$ after passing through a distance D across the region with nonnegligible friction.
• At time $t = t_{1}$, Block A is at x = $t_{3}$, and Block A collides with and sticks to Block B.
(a) For parts (a)(i) and (a)(ii), express your answer in terms of m, k , D, m, xc, and physical constants, as appropriate.

i. Derive an expression for the speed v of Block A at time $t_{1}$.

ii. Derive an expression for the speed vA,B of the two-block system immediately after the collision at time $t_{3}$.

(b) i. On the following axes, sketch a graph of the kinetic energy K of Block A as a function of time t from
time $t = 0$ to time $t_{3}$.

ii. Use principles of work and energy to justify the graph drawn in part (b)(i) for the time interval t = 0 to t = t1.

Explicitly reference features of the shape of the graph you drew in part (b)(i).

After the collision, the two-block system instantaneously comes to rest at time $t_{1}$, which occurs when the string
makes a small angle θmax with the vertical, as shown in Figure 2. For times $t > t_{1}$, the system oscillates with frequency f.

 The support holding the string is raised, and the procedure is then repeated using a new string of length 2.
(c) Indicate how the new frequency of oscillation f2 of the system on the new string of length 2 will compare to the frequency of oscillation f from the original procedure.
_____ f > f 2 _____ f < f 2 _____ f = f 2
Briefly justify your answer.

▶️Answer/Explanation

Ans:-

(a)(i) For a multi-step derivation with an application of the conservation of mechanical energy that indicates that all of the energy of the system is initially Us.

$E_{initial}=E_{final}$
$U_{s} = K$
$\frac{1}{2} kx_{c}^{2}=\frac{1}{2}mv^{2}$
$v=\sqrt{\frac{kx_{c}^{2}}{m}}$
$v=x_{c} \sqrt{\frac{k}{m}}$

(a)(ii) For a derivation to solve for the speed at $x_{2}$, that includes one of the following:
• An appropriate application of the conservation of energy
• An appropriate kinematics equation.

For one of the following that is consistent with the previous point in the response for part (a)(ii):

• A correct expression for the energy dissipated by friction
• A correct expression for the acceleration of the block in the region with nonnegligible friction

For attempting to derive an expression for $v_{A,B}$ by using the conservation of momentum

For substituting the expression for the speed at $x_{2}$ that is consistent with the first point of
the response in part (a)(ii) and substituting the correct masses into an expression for conservation of momentum

(b)(i) For a nonlinear sketch that begins at zero and increases for the entire time interval  $0 ≤ t ≤t_{1}$
For a sketch that decreases for the entire time interval $t_{1} ≤ t  ≤ t_{2}$ but does not go to zero 
For a sketch that is concave up for the time interval t_{1}≤ t≤t_{2}
For a continuous function for the time interval $t_{1} ≤ t ≤ t_{3} $that has a horizontal line that is greater than zero for the time interval $t_{2}≤t ≤t_{3}$.

(b)(ii) For a statement about the change in kinetic energy that is consistent with the graph drawn in the response for part (b)(i)

Example Response
From $0< t<t_{1}$, the kinetic energy of Block A increases. The force exerted on the block by
the compressed spring transfers the elastic potential energy in the block-spring system to the
kinetic energy of the block. Because the force exerted by the spring is not applied at
a constant rate, the kinetic energy of the block does not increase at a constant rate.
(c) For selecting  with an attempt at a relevant justification ,
For correctly applying an equation that relates the length of a pendulum to the period or frequency of the pendulum

Example Response
The period of a pendulum is calculated by using $T=2\pi \sqrt{\frac{l}{g}}$. Therefore, as the length is
increased, the period will also increase. Because frequency and period are inversely related,
an increase in period will result in a decrease in frequency.

Question

A small block of mass m starts from rest at the top of a frictionless ramp, which is at a height h above a horizontal tabletop, as shown in the side view above. The block slides down the smooth ramp and reaches point P with a speed v0 . After the block reaches point P at the bottom of the ramp, it slides on the tabletop guided by a circular vertical wall with radius R, as shown in the top view. The tabletop has negligible friction, and the coefficient of kinetic friction between the block and the circular wall is μ .
(a) Derive an expression for the height of the ramp h. Express your answer in terms of v0 , m, and fundamental constants, as appropriate.
A short time after passing point P, the block is in contact with the wall and moves with a speed of v .
(b)
i. Is the vertical component of the net force on the block upward, downward, or zero?
____ Upward ____ Downward ____ Zero
Justify your answer.
ii. On the figure below, draw an arrow starting on the block to indicate the direction of the horizontal component of the net force on the moving block when it is at the position shown.

Justify your answer.
Express your answers to the following in terms of v0 , u , m, R, m , and fundamental constants, as appropriate.
(c) Determine an expression for the magnitude of the normal force N exerted on the block by the circular wall as a function of v .
(d) Derive an expression for the magnitude of the tangential acceleration of the block at the instant the block has attained a speed of v .
(e) Derive an expression for v(t) , the speed of the block as a function of time t after passing point P on the track.

Answer/Explanation

Ans:

(a)

WNC = ΔA + Δv

0 = \(\frac{1}{2}mv^{2}-mgh\)

\(\frac{1}{2}{mv_{0}}^{2}= mgh\)

\(\therefore h = \frac{{v_{0}}^{2}}{2g}\)

(b) i.

  √    Zero

The block shows centripetal acceleration, but ∑F1 = 0 because it does not accelerate up or down. Thus, ms – N = 0.

ii.

The block is in circular motion about the center of  the loop. Thus, it must accelerate toward the center of the loop. Furthermore, friction with the loop’s wall provides a regarding force (tangible acceleration). These 2 forces together provide the vector shown above.

(c) \(F_{c} = \frac{mv^{2}}{r}\rightarrow \therefore N = \frac{mv^{2}}{R}\)

(d) 

\(\sum \overrightarrow{F}= ma\)

-μN = ma

\(-\mu \left ( \frac{mv^{2}}{R} \right )=ma\)

\(\therefore a = -\frac{\mu v^{2}}{R}\)

(e)

\(\frac{dv}{dt}= – \frac{\mu v^{2}}{R}\)

\(\int \frac{dv}{v^{2}}= \int -\frac{\mu }{R}dt\)

\(-\frac{1}{v}= -\frac{\mu t}{R}+C_{2}\)

When t = 0, v = v0

\(\therefore -\frac{1}{v_{0}}= 0 + c_{1}\)

\(\therefore c_{1}= -\frac{1}{v_{0}}\)

\(\therefore \frac{1}{v}= \frac{\mu t}{R}+\frac{1}{v_{0}}\)

\(\frac{1}{v}= \frac{\mu tv_{0}+R}{Rv_{0}}\)

\(\therefore v(t)= \frac{Rv_{0}}{\mu v_{0}t+R}\)

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