In the figure, gives the one-dimensional potential energy well for a $4.0 \mathrm{kg}$ particle (the function $U\left(x\right)$ has the form $b{x}^2$ and the vertical axis scale is set by $U_{\mathrm{s}}=2.0 \mathrm{J}$). $\left(a\right)$ If the particle passes through the equilibrium position with a velocity of $85 \mathrm{cm}/\mathrm{s}$, will it be turned back before it reaches $x=15 \mathrm{cm}$? $\left(b\right)$ If yes, at what position, and if no, what is the speed of the particle at $x=15 \mathrm{cm}$?

 

In the figure, a $5.00 \mathrm{kg}$ disk of diameter $\mathrm{D}=42.0 \mathrm{cm}$ is supported by a rod of length $\mathrm{L}=76.0 \mathrm{cm}$ and negligible mass that is pivoted at its end. (a) With the massless torsion spring unconnected, what is the period of oscillation? (b) With the torsion spring connected, the rod is vertical at equilibrium. What is the torsion constant of the spring if the period of oscillation has been decreased by $0.500 \mathrm{s}$?

In the figure, the pendulum consists of a uniform disk with radius, $r=10.0 \mathrm{cm}$ and mass $500 \mathrm{g}$ attached to a uniform rod with length $L=500 \mathrm{mm}$ and mass $250 \mathrm{g}$.
(a) Calculate the rotational inertia of the pendulum about the pivot point. (b) What is the distance between the pivot point and the center of mass of the pendulum? (c) Calculate the period of oscillation.

A physical pendulum has a centre of oscillation at distance $2L/3$ from its point of suspension. Show that the distance between the point of suspension and the centre of oscillation for a physical pendulum of any form is $\frac{\mathit{I}}{\mathit{m}\mathit{h}}$, where $I$ is moment of inertia, $h$ is distance of centre of mass from the pivot point and $m$ is the mass of the pendulum.

An oscillator consists of a block of mass $0.500 \mathrm{kg}$ connected to a spring. When set into oscillation with amplitude $35.0 \mathrm{cm}$, the oscillator repeats its motion every $0.350 \mathrm{s}$. Find the

$\left(a\right)$ period,

$\left(b\right)$ frequency,

$\left(c\right)$ angular frequency,

$\left(d\right)$ spring constant,

$\left(e\right)$ maximum speed, and

$\left(f\right)$ the magnitude of the maximum force on the block from the spring.

A $5.00 \mathrm{kg}$ disk of diameter $D=42.0 \mathrm{cm}$ is supported by a rod of length $76.0 \mathrm{cm}$ and negligible mass that is pivoted at its end.

$\left(a\right)$ With the massless torsion spring unconnected, what is the period of oscillation?

$\left(b\right)$ With the torsion spring connected, the rod is vertical at equilibrium. What is the torsion constant of the spring, if the period of oscillation has been decreased by $0.500 \mathrm{s}$?

The figure shows a physical pendulum consisting of a uniform disk with radius $r=10.0 \mathrm{cm}$ and mass $500 \mathrm{g}$ attached to a uniform rod with the length, $L=500 \mathrm{mm}$ and mass $250 \mathrm{g}$.

$\left(a\right)$ Calculate the rotational inertia of the pendulum about the pivot point/hinge.

$\left(b\right)$ What is the distance between the pivot point and the centre of mass of the pendulum?

$\left(c\right)$ Calculate the period of oscillation for small amplitude oscillations.

The suspension system of a $2400 \mathrm{kg}$ automobile sags $10 \mathrm{cm}$ when the chassis is placed on it. Also, the oscillation amplitude decreases by $50\%$ per cycle. Estimate the values of,

(a) the spring constant $k$, and

(b) the damping constant $b$ for the spring and shock absorber system of one wheel, assuming each wheel supports $600 \mathrm{kg}$.