Oscillations

1-1

The initial position and initial velocity of an object moving in simple harmonic motion are $x_i$ and $v_i$; the angular frequency of oscillation is $\omega$ . (a) show that the position and velocity of the object for all time can be written as

$$x(t)=x_i\cos\omega t+ \left({v_i\over \omega}\right) \sin \omega t$$

$$v(t) = - x_i\omega\sin \omega t + v_i\cos \omega t$$

. (b) If the amplitude of the motion is $A$, show that

$$v^2 -ax = v_i^2-a_ix_i = \omega^2A^2$$

.

1-2

The end of one of the prongs of a tuning fork that executes simple harmonic motion of frequency 2000 Hz has an amplitude of 0.40 mm. Find (a) the maximum acceleration and maximum speed of the end of the prong (b) acceleration and speed of the end of the prong when it has a displacement .02 mm.

1-3

The Body Mass Measurement Device (BMMD) is designed for use on orbiting space vehicles, its purpose is to allow astronauts to measure their mass in the weightless condition in earth orbit. The BMMD is a spring-mounted chair; an astronaut measures his or her period of oscillation in the chair; the mass follows from the formula for the period of an oscillating block-spring system. (a) If mass $M$ is the mass of the astronaut and $m$ is the effective mass of that part of the BMMD that also oscillates, show that

$$M=(k/4\pi^2)T^2 -m ,$$

where $T$ is the period of oscillation and $k$ is the spring constant. (b) The spring constant is $k= 605.6 \rm N/m$ for the BMMD on Skylab Mission Two; the period of oscillation of the empty chair is 0.90149 s. Calculate the effective mass of the chair. (c) With an astronaut in the chair, the period of oscillation becomes 2.08832 s. Calculate the mass of the astronaut.

1-4

An automobile can be considered to be mounted on four springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the vibrations have a frequency of 3.0 Hz (a) What is the force constant of each of the four springs (assumed identical) if the mass of the car is 1450 kg? (b) What will be the vibration frequency be if five passengers, averaging 73 kg each, ride in the car?

1-5

An oscillator consists of a block attached to a spring ($k=400 \rm N/m$). At some time $t$, the position (measured from the equilibrium location), velocity, and acceleration of the block are $x=0.10 \rm m$, $v=-13.6 \rm m/s$, $a=-123 \rm m/s^2$. Calculate (a) the frequency, (b) the mass of the block, and (c) the amplitude of oscillation.

1-6

An oscillating block-spring system has a mechanical energy of 1.0 J, an amplitude of 0.10 m, and a maximum speed of 1.2 m/s. Find (a) the force constant of the spring, (b) the mass, and (c) the frequency of oscillation.

1-7

A 2.0 kg mass is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of 20.0 N is required to hold the mass at rest when it is pulled 0.200 m from its equilibrium position ( the origin of the $x$ axis). The mass is now released from rest with an initial displacement of $x_i =0.200 \rm m$, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the mass. Where does this maximum speed occur? (d) Find the maximum acceleration of the mass. Where does it occur? (e) Find the total energy of the oscillating system. Find (f) the speed and (g) the acceleration when the displacement equals one third of the maximum value.

1-8

A pendulum is formed by pivoting a long thin rod of length $L$ and mass $m$ about a point on the rod that is a distance $d$ above the centre of the rod. Find the small amplitude period of this pendulum in terms of $d$, $L$, $m$, and $g$.

1-9

A simple pendulum is 5.00 m long. (a) What is the period of simple harmonic motion for this pendulum if it is hanging in an elevator that is accelerating upward at $5.00 \rm m/s^2$? (b) What is its period if the elevator is accelerating downward at $5.00 \rm m/s^2$? (c) What is the period of simple harmonic motion for this pendulum if it is placed in truck that is accelerating horizontally at $5.00 \rm m/s^2$?

1-10

A baby rejoices in the day by crowing and jumping up and down in her crib. Her mass is 12.5 kg, and the crib mattress can be modelled as light spring with a force constant of 4.30 kN/m. (a) The baby soon learns to bounce with maximum amplitude and minimum effort by bending her knees at what frequency? (b) She learns to use the mattress as a trampoline-losing contact with it for part of each cycle-when her amplitude exceeds what value?