15.6 Forced Oscillations

A famous magic trick involves a performer singing a note toward a crystal glass until the glass shatters. Explain why the trick works in terms of resonance and natural frequency.

Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?

If a car has a suspension system with a force constant of $5.00 \times 10^{4} \text{N}/\text{m}$, how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?

Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction $\mu_{\text{s}} = 0.100$. (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coeff…

A diver on a diving board is undergoing SHM. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver's period of simple harmonic oscillation is 1.05 s. What is the second diver's mass if the mass of the board is negligible?

Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is $1.63 \text{m}/\text{s}^{\text{2}}$.

If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g object is fired with a speed of 20 m/s at the 2.00-kg object, and the two objects collide and stick together in a totally inelastic collision. Write an equation for the motion of the system after the collision. Assume air resistance is negligible.

A 2.00-kg block lies at rest on a frictionless table. A spring, with a spring constant of 100 N/m is attached to the wall and to the block. A second block of 0.50 kg is placed on top of the first block. The 2.00-kg block is gently pulled to a position $x = + A$ and released from rest. There is a coefficient of friction of 0.45 between the two blocks. (a) What is the period of the oscillations? (b) What is…

Near the top of the Citigroup Center building in New York City, there is an object with mass of $4.00 \times 10^{5} \text{kg}$ on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective f…

Consider the van der Waals potential $U \left(r\right) = U_{o} \left[\left(\frac{R_{o}}{r}\right)^{12} - 2 \left(\frac{R_{o}}{r}\right)^{6}\right]$, used to model the potential energy function of two molecules, where the minimum potential is at $r = R_{o}$. Find the force as a function of r. Consider a small displacement $r = R_{o} + r^{'}$ and use the binomial theorem:

$\left(1 + x\right)^{n} = 1 + n x + \frac{n \left(n - 1\right)}{2 !} x^{2} + \frac{n \left(n - 1\right) \left(n - 2\right)}{3 !} x^{3} + \hdots$,

to show that the force does approximate a Hooke’s law force.

(a) The springs of a pickup truck act like a single spring with a force constant of $1.30 \times 10^{5} \text{N}/\text{m}$. By how much will the truck be depressed by its maximum load of 1000 kg? (b) If the pickup truck has four identical springs, what is the force constant of each?