A block oscillating on a spring has period $T=2\text{s}$. What is the period if:

1. The block’s mass is doubled? Explain. Note that you do not know the value of either m or k, so do not assume any particular values for them. The required analysis involves thinking about ratios.

2. The value of the spring constant is quadrupled?

3. The oscillation amplitude is doubled while m and k are unchanged?

A pendulum on Planet X, where the value of g is unknown, oscillates with a period $T=2\text{s}$. What is the period of this pendulum if:

1. Its mass is doubled? Explain. Note that you do not know the value of m, L, or g, so do not assume any specific values. The required analysis involves thinking about ratios.

2. Its length is doubled?

3. Its oscillation amplitude is doubled?

FIGURE Q15.3 shows a position-versus-time graph for a particle in SHM. What are (a) the amplitude A, (b) the angular frequency $\omega$, and (c) the phase constant ${\varphi }_0?$

FIGURE Q15.3

FIGURE Q15.4 shows a position-versus-time graph for a particle in SHM.

1. What is the phase constant ${\varphi }_0?$ Explain.

2. What is the phase of the particle at each of the three numbered points on the graph?

FIGURE Q15.4

Equation 15.25 states that $\frac{1}{2}k{A}^2=\frac{1}{2}m(v_{\text{max}}{)}^2$. What does this mean? Write a couple of sentences explaining how to interpret this equation.

A block oscillating on a spring has an amplitude of 20 cm. What will the amplitude be if the total energy is doubled? Explain.

A block oscillating on a spring has a maximum speed of $20\text{cm}\text{/}\text{s}$. What will the block’s maximum speed be if the total energy is doubled? Explain.

The solid disk and circular hoop in FIGURE Q15.8 have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at the top edge. Which, if either, has the larger period of oscillation?

FIGURE Q15.8

FIGURE Q15.9 shows the potential-energy diagram and the total energy line of a particle oscillating on a spring. The left end of the spring is fixed at $x=0\text{cm}$.

1. What is the spring’s equilibrium length?

2. Where are the turning points of the motion? Explain.

3. What is the particle’s maximum kinetic energy?

4. What will be the turning points if the particle’s total energy is doubled?

FIGURE Q15.9

Suppose the damping constant b of an oscillator increases.

1. Is the medium more resistive or less resistive?

2. Do the oscillations damp out more quickly or less quickly?

3. Is the time constant $\tau$ increased or decreased?

1. Describe the difference between $\tau$ and T. Don’t just name them; say what is different about the physical concepts they represent.

2. Describe the difference between $\tau$ and $t_{1\text{/}2}$.

What is the difference between the driving frequency and the natural frequency of an oscillator?