### Problem 1

#### Description

**A 0.5000-kg cart resting on an air track oscillates as shown in the animation (position in meters
and time in seconds). NO FRICTION !**

#### QUESTION 1

**What is the spring constant k of the spring?
Start**

[REQUIREMENT: Use conservation of energy. Set the maximum potential energy
equal to the maximum kinetic energy.

You can get the maximum KE from an *approximation* of the speed at x = 0 by stepping through
the origin
with an increment on either side to get the change in position and using
other pertinent information.]

QUESTION 2

In Chapter 14, we will discover the position x may be written under
special initial conditions as x = A*cosωt,
where A is the amplitude or magnitude of the maximum displacement from x = 0
and ω is the angular frequency. The expression
means the mass on the spring was released from rest at initial x = A.
You may verify this by finding the velocity. Differentiate the formula
for x. From this information, answer and explain the following question.
The method of computation in #1 will produce a systematic error such that the
calculated k above will differ from the true value. Will
computed k be larger or smaller than the true value? Explain graphically
with a sketch.

**QUESTION 3 **

**What is the value of positive x such that the kinetic energy equals the
potential energy ? Start**

Find this using a calculation. How does your x calculation compare
with the x-value inferred from the graph?

Explain.

**Reference: CH. 6, 7**