Show that when a metal rod $L$ meters long moves at speed $v$ perpendicular to $\vec{B}$ field lines, the magnetic force exerted by the field on the electrically charged particles in the rod produces a potential difference between the ends of the rod equal to the product $B L v$.

Determine a mass density $\rho$ so that the center of mass of a rod of length L will be at a distance L/4 from one end of the rod.

A thin uniform rod of length l and mass m is constrained to rotate with constant angular velocity $\omega$ about an axis passing through the center O of the rod and making an angle ${\alpha}$ with the rod. (a) Show that the angular momentum L about O is perpendicular to the rod and is of magnitude $\left(m l^{2} \omega / 12\right) \sin \alpha$. (b) Show that the torque vector N is perpendicular to the rod and to L and is of magnitude $\left(m l^{2} \omega^{2} / 12\right) \sin \alpha \cos \alpha$.

$M 1$ and $M 2$ masses are placed at opposite ends of a rigid rod of length $L$ and negligible mass; the dimensions of $M_1$ and $M_2$ are negligible in comparison to $L$. The rod should be rotated around an axis perpendicular to it. What point $x$ on this rod (measured from $M_2$) should the axis pass through in order to minimise the work required to set the rod rotating with an angular speed $\omega 0$?

At 20°C, a rod is exactly 20.05 cm long on a steel ruler. Both are placed in an oven at 270°C, where the rod now measures 20.11 cm on the same ruler. What is the coefficient of linear expansion for the material of which the rod is made?

Give density functions of thin rod lying along various interval of the x-axis. Use Equations (3a) through (3c) to find the rod's moment about the origin, mass, and center of mass.

$$\delta(x)= \begin{cases}2-x, & 0 \leq x<1 \\ x, & 1 \leq x \leq 2\end{cases}$$

A compression member of $20 -\text{in.}$ effective length consists of a solid $1-\text{in.}$-diameter aluminum rod. In order to reduce the weight of the member by 25%, the solid rod is replaced by a hollow rod of the cross section shown. Determine (a) the percent reduction in the critical load, (b) the value of the critical load for the hollow rod. Use $E=10.6 \times 10^6 \mathrm{psi}$

When a glass rod is rubbed with silk, the rod becomes positively charged, but when a rubber rod is rubbed with fur, the rubber becomes negatively charged. Suppose you have a charged object but don't know whether its charge is positive or negative. Explain how you could use a glass rod and piece of silk to determine the sign of the unknown charge on the object.

The uniform rod with center of mass at G is supported by the pegs A and B, which are fixed in the wheel. If the coefficient of friction between the rod and pegs is $\mu$, determine the angle $\theta$ through which the wheel may be slowly turned about its horizontal axis through O, starting from the position shown, before the rod begins to slip. Neglect the diameter of the rod compared with the other dimensions.

A uniform 0.25-kg thin rod that has an 80-cm length is free to rotate about a fixed horizontal axis perpendicular to and through one end of the rod. It is held horizontal and released. Immediately after it is released, what is

$(a)$ the acceleration of the center of the rod?

A collar with a mass of $1 \mathrm{~kg}$ is rigidly attached at a distance $d=$ $300 \mathrm{~mm}$ from the end of a uniform slender rod $A B$. The rod has a mass of $3 \mathrm{~kg}$ and is of length $L=600 \mathrm{~mm}$. Knowing that the rod is released from rest in the position shown, find the angular velocity of the rod after it has rotated through $90^{\circ}$.

One end of a long glass rod (n = 1.5) is a convex surface of radius 6.0 cm. An object is located in air along the axis of the rod, at a distance of 10 cm from the convex end. (a) How far apart are the object and the image formed by the glass rod? (b) Within what range of distances from the end of the rod must the object be located in order to produce a virtual image?

A rod extends from $x=0$ to $x=a, a>0$. Find the center of mass if the density of the rod varies directly as the distance from $x=\frac{1}{4}a$.

A rod is pivoted about one end. Two forces $\overrightarrow{\mathbf{F}}$ and $-\overrightarrow{\mathbf{F}}$ are applied to it. Under what circumstances will the rod not rotate?