Solving Harmonic Oscillator Problems

(See attached file for full problem description)

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Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions.
A block of mass m is attached to a spring whose spring constant is k. The other end of the spring is fixed so that when the spring is unstretched, the mass is located at x=0. Assume that the +x direction is to the right.
The mass is now pulled to the right a distance A beyond the equilibrium position and released, at time t=0 with zero initial velocity.

Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is (see attached file), and its solution, which provides the equation for x(t), is (see attached file).

A) At what time t1 does the block come back to its original equilibrium position (x=0) for the first time?
Express your answer in terms of k and m.
t1 =

B) Find the velocity v of the block as a function of time.
Express your answer in terms of k, m, A, and t.
v(t) =

C) Find the acceleration a of the block as a function of time.
Express your answer in terms of k, m, A, and t.
a(t) =

D) Specify when the magnitude of the acceleration of the block reaches its maximum value. Consider the following options:
a. only once during one period of motion,
b. when the block's speed is zero,
c. when the block is in the equilibrium position,
d. when the block's displacement equals either A or -A,
e. when the block's speed is at a maximum.
Choose the most complete answer.

a only

b only

c only

d only

e only

b and d

c and e

b and c

a and e

d and e

E) Find the kinetic energy K of the block as a function of time.
Express your answer in terms of k, m, A, and t.
K(t) =

F) Find kmax, the maximum kinetic energy of the block.
Express your answer in terms of k, m, and A.
Kmax =

G) The kinetic energy of the block reaches its maximum when which of the following occurs?

The displacement of the block is zero.

The displacement of the block is A.

The acceleration of the block is at a maximum.

The velocity of the block is zero.
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