A child's toy consists of a spherical object of mass 50 g attached to a spring. One end of the spring is fixed to the side of the baby's crib so that when the baby pulls on the toy and lets go, the object oscillates horizontally with a simple harmonic motion. The amplitude of the oscillation is 6 cm and the maximum velocity achieved by the toy is 3.2 m/s . What is the kinetic energy K of the toy when the spring is compressed 5.1 cm from its equilibrium position?Problem-Solving Strategy: Simple Harmonic Motion II: Energy

The energy equation, E=12mvx2+12kx2=12kA2, is a useful alternative relationship between velocity and position, especially when energy quantities are also required. If the problem involves a relationship among position, velocity, and acceleration without reference to time, it is usually easier to use the equation for simple harmonic motion, ax=d2xdt2=−kmx (from Newton’s second law) or the energy equation above (from energy conservation) than to use the general expressions for x, vx, and ax as functions of time. Because the energy equation involves x2 and vx2, it cannot tell you the sign of x or of vx; you have to infer the sign from the situation. For instance, if the body is moving from the equilibrium position toward the point of greatest positive displacement, then x is positive and vx is positive.

IDENTIFY the relevant concepts

Energy quantities are required in this problem, therefore it is appropriate to use the energy equation for simple harmonic motion.

SET UP the problem using the following steps

Part A

The following is a list of quantities that describe specific properties of the toy. Identify which of these quantities are known in this problem.

Select all that apply.

Select all that apply.

maximum velocity vmax
amplitude A
force constant k
mass m
total energy E
potential energy U at x
kinetic energy K at x
position x from equilibrium

Part B

What is the kinetic energy of the object on the spring when the spring is compressed 5.1 cm from its equilibrium position?

Part C

What is the potential energy U of the toy when the spring is compressed 5.1 cm from its equilibrium position?

1
Verify the identity. Show your work.

cot θ ∙ sec θ = csc θ

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