Additional Problems

Suppose you attach an object with mass m to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length, the length of the spring in equilibrium, without the mass attached. The amplitude of the motion is the distance between the equilibrium position of the spring without the mass attached and the equilibrium position of the spring with the mass attached. (a) Show that the spring exerts an upward force of 2.00mg on the object at its lowest point. (b) If the spring has a force constant of 10.0 N/m, is hung horizontally, and the position of the free end of the spring is marked as $y=0.00\text{m}$, where is the new equilibrium position if a 0.25-kg-mass object is hung from the spring? (c) If the spring has a force constant of 10.0 N/m and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (d) Find the maximum velocity.

Suppose a diving board with no one on it bounces up and down in a SHM with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the SHM of a 75.0-kg diver on the board?

A mass is placed on a frictionless, horizontal table. A spring $\left(k=100\text{N/m}\right)$, which can be stretched or compressed, is placed on the table. A 5.00-kg mass is attached to one end of the spring, the other end is anchored to the wall. The equilibrium position is marked at zero. A student moves the mass out to $x=4.00\text{cm}$ and releases it from rest. The mass oscillates in SHM. (a) Determine the equations of motion. (b) Find the position, velocity, and acceleration of the mass at time $t=3.00\text{s}\text{.}$

At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is $1.63{\text{m/s}}^{\text{2}}$, if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.

A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g mass is fired with a speed of 20 m/s at the 2.00-kg mass, and the 100.00-g mass collides perfectly elastically with the 2.00-kg mass. Write an equation for the motion of the hanging mass after the collision. Assume air resistance is negligible.

Assume that a pendulum used to drive a grandfather clock has a length $L_0=1.00\text{m}$ and a mass M at temperature $T=20.00\text{°}\text{C}\text{.}$ It can be modeled as a physical pendulum as a rod oscillating around one end. By what percentage will the period change if the temperature increases by $10\text{°}\text{C}?$ Assume the length of the rod changes linearly with temperature, where $L=L_0\left(1+\alpha \text{Δ}T\right)$ and the rod is made of brass $\left(\alpha =18\times {10}^{\mathrm{-6}}\text{°}{\text{C}}^{\mathrm{-1}}\right).$