Representing and Analyzing SHM AP Physics C Mechanics FRQ – Exam Style Questions etc.
Representing and Analyzing SHM AP Physics C Mechanics FRQ
Unit 7: Oscillations
Weightage : 10-15%
Question
A uniform rod of length d has one end fixed to the central axis of a horizontal, frictionless circular platform of radius R = 2d . Fixed at the other end of the rod is an ideal spring of negligible mass to which a block is attached. The block is set in frictionless grooves so that it can only move along a radius of the platform, as shown in Figure 1 above. The equilibrium length of the spring is d/2 . Below is a table showing the mass of the block and the masses and rotational inertias of the rod and platform.
A motor begins to slowly rotate the platform counterclockwise as viewed from above until the platform reaches a constant angular speed ω . Under these conditions, the spring has stretched by an additional length d/2 , as shown in Figure 2.
Answer the following questions for the platform rotating at constant angular speed ω . Express all algebraic answers in terms of m, d, ω , and physical constants, as appropriate.
(a) Derive an expression for the spring constant of the spring.
(b)
i. Determine an expression for the rotational inertia of the block around the axis of the platform.
ii. Derive an expression for the rotational inertia of the entire system about the axis of the platform.
(c) Determine an expression for the angular momentum of the entire system about the axis of the platform
While the system continues to rotate, a small mechanism in the pivot moves the rod slowly until the center of the rod is positioned on the axis, as shown in Figure 3 above. The same constant angular speed ω is maintained by the motor driving the platform.
(d) Derive an expression for the distance x that the spring is stretched when the rod reaches the position shown in Figure 3 above.
For parts (e), (f), and (g), assume the center of the rod is still moving toward the axis of the platform.
(e) Is the angular momentum of the entire system increasing, decreasing, or staying the same?
_____Increasing _____Decreasing _____Staying the same
Justify your answer.
(f) In order to keep the system rotating with constant angular speed ω , is the motor doing positive work, negative work, or no work on the rotating system?
_____Positive _____Negative _____No work
Justify your answer.
(g) On the block in Figure 4 below, draw a single vector representing the direction of the acceleration of the block. Draw the vector so that it is starting on, and pointing away from, the block.
Answer/Explanation
Ans:
(a)
Since Fspring = Fcentripetal,
\(\frac{kd}{2}= mw^{2}(d + \frac{d}{2}+\frac{d}{2})\)
So \(\frac{kd}{2}= 2mdw^{2}\)
So k = 4mw2
(b) i.
\(I = mr^{2}= m(d + \frac{d}{2}+\frac{d}{2})^{2}=4md^{2}\)
ii.
Isys = Iblock + I rod + Iplatform
\(= 4md^{2}+\frac{3md^{2}}{3}+\frac{5m(2d)^{2}}{2}\)
= 4md2 + md2 + 10md2
= 15 md2
(c)
L = Iw = 15 md2w
(d)
Since Fspring = Fcentripetal,
\(kx = mw^{2}\left ( \frac{d}{2}+\frac{d}{2}+x \right )\)
So 4mw2x = mw2 (d+x)
So d + x = 4x
So 3x = d
So X = d/3
(e)
√ Decreasing
By parallel – axis thebrem, we know that the closer the center of mass of an objet is to the rotational axis, the lower the rotational inertia is. Since the rod takes the rod and the block closer to the center, their intertia decreases. Since w stay the same, L = Iw will decrease.
(f)
√ Negative
If there were no motor, by conservation of angular momentum, Since I decreased, w will increase. So in order to maintain constant w, the motor have to do negative workto slow down the angular speed.
(g)
