Energy of Simple Harmonic Oscillators AP Physics C Mechanics FRQ – Exam Style Questions etc.
Energy of Simple Harmonic Oscillators AP Physics C Mechanics FRQ
Unit 7: Oscillations
Weightage : 20-15%
Question
A block of mass 2M rests on a horizontal, frictionless table and is attached to a relaxed spring, as shown in the figure above. The spring is nonlinear and exerts a force f(x) = – Bx3, where B is a positive constant and x is the displacement from equilibrium for the spring. A block of mass 3M and initial speed v0 is moving to the left as shown.
(a) On the dots below, which represent the blocks of mass 2M and 3M, draw and label the forces (not components) that act on each block before they collide. Each force must be represented by a distinct arrow starting on, and pointing away from, the appropriate dot.
The two blocks collide and stick to each other. The two-block system then compresses the spring a maximum distance D, as shown above. Express your answers to parts (b), (c), and (d) in terms of M, B, v0 , and physical constants, as appropriate.
(b) Derive an expression for the speed of the blocks immediately after the collision.
(c) Determine an expression for the kinetic energy of the two-block system immediately after the collision.
(d) Derive an expression for the maximum distance D that the spring is compressed.
(e)
i. In which direction is the net force, if any, on the block of mass 2M when the spring is at maximum compression?
____ Left ____Right ____ The net force on the block of mass 2M is zero.
Justify your answer.
ii. Which of the following correctly describes the magnitude of the net force on each of the two blocks when the spring is at maximum compression?
____ The magnitude of the net force is greater on the block of mass 2M.
____ The magnitude of the net force is greater on the block of mass 3M.
____ The magnitude of the net force on each block has the same nonzero value.
____ The magnitude of the net force on each block is zero.
Justify your answer.
____
____
(f) Do the two blocks, which remain stuck together and attached to the spring, exhibit simple harmonic motion after the collision?
____ Yes ____ No
Justify your answer.
Answer/Explanation
Ans:
(a)
(b)
3MV0 = 5MV
\(\frac{3}{5}V_{0} = V\)
(c)
\(KE = \frac{1}{2}MV^{2}\)
\(= \frac{1}{2}SM\left ( \frac{3}{5}V_{0} \right )^{2}\)
\(KE = \frac{9}{10}M{V_{0}}^{2}\)
(d)
\(W = \int -Bx^{3}dx = \frac{Bx^{4}}{4}\)
\(\frac{9}{10}M{V_{0}}^{2} = \frac{BD^{4}}{4}\)
\(\sqrt[4]{\frac{3.6M{V_{0}}^{2}}{B}}= D\)
(e) i.
√ Right
The block will start accelerating to the right due to the force of the spring of the maximum distance, so the net force is to the right.
ii.
√ The magnitude of the net force is greater on the block of mass 3M.
The acceleration is constant, but the net force on the 3M block will be greater since 3Ma > 2Ma
(f)
√ No
Sma the surface is frictionless, while the 2M block will follow SMH, the 3M block will continue to the right with a constant acceleration