A 500 g block slides along a frictionless surface at a speed...

A 500 g block slides along a frictionless surface at a speed of $0.35 \mathrm{m} / \mathrm{s} .$ It runs into a horizontal massless spring with spring constant $50 \mathrm{N} / \mathrm{m}$ that extends outward from a wall. It compresses the spring, then is pushed back in the opposite direction by the spring, eventually losing contact with the spring. a. How long does the block remain in contact with the spring? b. How would your answer to part a change if the block's initial speed were doubled?

A 500 g block slides along a frictionless surface at a speed of $0.35 \mathrm{m} / \mathrm{s} .$ It runs into a horizontal massless spring with spring constant $50 \mathrm{N} / \mathrm{m}$ that extends outward from a wall. It compresses the spring, then is pushed back in the opposite direction by the spring, eventually losing contact with the spring.
a. How long does the block remain in contact with the spring?
b. How would your answer to part a change if the block's initial speed were doubled?

Key Concepts

Conservation of Energy

This concept involves recognizing that in a frictionless system, the total mechanical energy (kinetic plus potential) remains constant. When the block compresses the spring, its kinetic energy is converted into the spring's potential energy and then converted back into kinetic energy as it is pushed back, ensuring energy equilibrium throughout the motion.

Simple Harmonic Motion (SHM)

The compression and decompression of the spring with the block attached can be modeled as simple harmonic motion. In SHM, the restoring force is proportional to the displacement from equilibrium, leading to sinusoidal behavior of the system with a predictable period determined by the mass and the spring constant.

Oscillation Period

For a mass-spring system undergoing simple harmonic motion, the period of oscillation is calculated using T = 2??(m/k). Understanding this period is essential because the time the block is in contact with the spring corresponds to a specific fraction of the full oscillation cycle.

Dependence on Initial Speed

In the context of simple harmonic motion, while the initial speed affects the amplitude of the motion (i.e., the maximum compression of the spring), it does not change the period of oscillation. Thus, even if the block's initial speed were doubled, the time contact with the spring, determined as a function of the oscillation period, would remain the same.

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