The disk rotates in the horizontal plane with constant angular velocity $\Omega=12 \mathrm{rad} / \mathrm{s}$. The mass $m=2 \mathrm{~kg}$ slides in a smooth slot in the disk and is attached to a spring with constant $k=860 \mathrm{~N} / \mathrm{m}$. The radial position of the mass when the spring is unstretched is $r=0.2 \mathrm{~m}$. At $t=0$, the mass is in the position $r=0.4 \mathrm{~m}$ and $d r / d t=0$. Determine the position $r$ as a function of time.

The astronaut's center of mass is at $x=1.01 \mathrm{~m}, y=0.16 \mathrm{~m}$, and his mass is $81.6 \mathrm{~kg}$. What is his moment of inertia about the $z^{\prime}$ axis through his center of mass?

Engineers use the device shown to measure an astronaut's moment of inertia. The horizontal board is pinned at $O$ and supported by the linear spring with constant $k=12 \mathrm{kN} / \mathrm{m}$. When the astronaut is not present, the frequency of small vibrations of the board about $O$ is measured and determined to be $6.0 \mathrm{~Hz}$. When the astronaut is lying on the board as shown, the frequency of small vibrations of the board about $O$ is $2.8 \mathrm{~Hz}$. What is the astronaut's moment of inertia about the $z$ axis?

A slender bar of mass $m$ and length $l$ is pinned to a fixed support as shown. A torsional spring of constant $k$ attached to the bar at the support is unstretched when the bar is vertical. Show that the equation governing small vibrations of the bar from its vertical equilibrium position is

$$\frac{d^2 \theta}{d t^2}+\omega^2 \theta=0, \quad \text { where } \omega^2=\frac{\left(k-\frac{1}{2} m g l\right)}{\frac{1}{3} m l^2} .$$

Consider the system described. At $t=0$, the angle $\beta=0.01$ rad and $d \beta / d t=0$. Determine $\beta$ as a function of time.

Suppose that the pulley has radius $R=4$ in and its moment of inertia is $I=0.06$ slug- $\mathrm{ft}^2$. The suspended object weighs $5 \mathrm{lb}$, and the spring constant is $k=10 \mathrm{lb} / \mathrm{ft}$. The system is initially at rest in its equilibrium position. At $t=0$, the suspended object is given a downward velocity of $1 \mathrm{ft} / \mathrm{s}$. Determine the position of the suspended object relative to its equilibrium position as a function of time.

The thin rectangular plate is attached to the rectangular frame by pins. The frame rotates with constant angular velocity $\omega_0=6 \mathrm{rad} / \mathrm{s}$. The angle $\beta$ between the $z$ axis of the body-fixed coordinate system and the vertical is governed by the equation

$$\frac{d^2 \beta}{d t^2}=-\omega_0^2 \sin \beta \cos \beta .$$

Determine the frequency of small vibrations of the plate relative to its horizontal position.

Strategy: By writing $\sin \beta$ and $\cos \beta$ in terms of their Taylor series and assuming that $\beta$ is small, show that the equation governing $\beta$ can be expressed.

The mass of the suspended object $A$ is $4 \mathrm{~kg}$. The mass of the pulley is $2 \mathrm{~kg}$ and its moment of inertia is $0.018 \mathrm{~N}-\mathrm{m}^2$. The spring constant is $k=150 \mathrm{~N} / \mathrm{m}$. The spring is unstretched when $x=0$. At $t=0$, the system is released from rest with $x=0$. What is the velocity of the object $A$ at $t=1 \mathrm{~s}$ ?

The mass of the suspended object $A$ is $4 \mathrm{~kg}$. The mass of the pulley is $2 \mathrm{~kg}$ and its moment of inertia is $0.018 \mathrm{~N}-\mathrm{m}^2$. The spring constant is $k=150 \mathrm{~N} / \mathrm{m}$. For vibration of the system relative to its equilibrium position, determine (a) the frequency in $\mathrm{Hz}$ and (b) the period.

The 2-lb bar is pinned to the 5-lb disk. The disk rolls on the circular surface. What is the frequency of small vibrations of the system relative to its vertical equilibrium position?

The 20 -lb disk rolls on the horizontal surface. Its radius is $R=6 \mathrm{in}$. The spring constant is $k=15 \mathrm{lb} / \mathrm{ft}$. At $t=0$, the spring is unstretched and the disk has a clockwise angular velocity of $2 \mathrm{rad} / \mathrm{s}$. What is the amplitude of the resulting vibrations of the center of the disk?

The 20-lb disk rolls on the horizontal surface. Its radius is $R=6$ in. Determine the spring constant $k$ so that the frequency of vibration of the system relative to its equilibrium position is $f=1 \mathrm{~Hz}$.

The spring constant is $k=800 \mathrm{~N} / \mathrm{m}$, and the spring is unstretched when $x=0$. The mass of each object is $30 \mathrm{~kg}$. The inclined surface is smooth. The radius of the pulley is $120 \mathrm{~mm}$ and its moment of inertia is $I=0.03 \mathrm{~kg}-\mathrm{m}^2$. At $t=0, x=0$ and $d x / d t=1 \mathrm{~m} / \mathrm{s}$. (a) Determine the frequency and period of the resulting vibration. (b) What is the value of $x$ at $t=4 \mathrm{~s}$ ?

The spring constant is $k=800 \mathrm{~N} / \mathrm{m}$, and the spring is unstretched when $x=0$. The mass of each object is $30 \mathrm{~kg}$. The inclined surface is smooth. Neglect the mass of the pulley. The system is released from rest with $x=0$. (a) Determine the frequency and period of the resulting vibration. (b) What is the value of $x$ at $t=4 \mathrm{~s}$ ?