A uniform plank of length $6.00 \mathrm{~m}$ and mass $30.0 \mathrm{~kg}$ rests horizontally across two horizontal bars of a scaffold. The bars are $4.50 \mathrm{~m}$ apart, and $1.50 \mathrm{~m}$ of the plank hangs over one side of the scaffold. Draw a free-body diagram of the plank. How far can a painter of mass $70.0 \mathrm{~kg}$ walk on the overhanging part of the plank before it tips?

The problem below concerns a rotating solid body and its moment of inertia about an axis; this moment relates angular acceleration to torque (an analogue of force). For a body of constant density and mass $m$ occupying a region W of volume V, the moments of inertia about the coordinate axes are

$$\begin{aligned} I_x & =\frac{m}{V} \int_W\left(y^2+z^2\right) d V \quad I_y=\frac{m}{V} \int_W\left(x^2+z^2\right) d V \\ I_z & =\frac{m}{V} \int_W\left(x^2+y^2\right) d V \end{aligned}$$

Find the moment of inertia about the x-axis of the rectangular solid $-a \leq x \leq a,-b \leq y \leq b$ and $-c \leq z \leq c$ of mass $m$.

The moment of inertia $I$ of a rigid body for an ax is through its center of mass is $I_{\mathrm{cm}}$. Is there any other axis parallel to this axis for which $I$ is less than $I_{\mathrm{rm}}$ ? Explain.

Earth was more than 4 billion years old before complex multicellular organisms with hard body parts appeared. Would you expect to find similar life forms on a planet orbiting a star with 10 times the sun's mass? Why?

Model the excavator's arm $\mathrm{ABC}$ as a single rigid body. Its mass is $1200 \mathrm{~kg}$, and the moment of inertia about its center of mass is $I=3600 \mathrm{~kg}-\mathrm{m}^2$. The angular velocity of the arm is $2 \mathrm{rad} / \mathrm{s}$ counterclockwise and its angular acceleration is $1 \mathrm{rad} / \mathrm{s}^2$ counterclockwise. What are the components of the force exerted on the arm at $A$ ?

Use the BMI indexes given below. Use a 0.01 significance level to test the claim that recent Miss America winners are from a population with a standard deviation equal to 1.34, which was the standard deviation of BMI for winners from the 1920s and 1930s. Do recent winners appear to have variation that is different from that of the 1920s and 1930s?

$$\begin{array} { c c c c c c c c c } { 19.5 } & { 20.3 } & { 19.6 } & { 20.2 } & { 17.8 } & { 17.9 } & { 19.1 } & { 19.8 } & { 17.6 } & { 16.8 } \end{array}$$

A body of mass $m$ moves along a trajectory $\vec{r}(t)$ in three-dimensional space with constant kinetic energy. What geometric relationship has to exist between the body's velocity vector, $\vec{v}(t)$, and its acceleration vector, $\vec{a}(t)$, in order to accomplish this?

In 0.181 s, through a distance of 8.05 m, a test pilot’s speed decreases from 88.9 m/s to 0 m/s. If the pilot’s mass is 70.0 kg, how much work is done against his body?

Wright Company makes 75 percent of its sales in cash. Credit sales are collected as follows: 65 percent in the month of sale and 35 percent in the month following the sale.

Wright's budgeted sales for upcoming months follow:

$$\begin{array}{lr}\text { June } & \$ 22,500 \\ \text { July } & 25,000 \\ \text { August } & 23,000 \\ \text { September } & 21,000\end{array}$$

Required:

Compute Wright’s expected cash receipts for August

In the given exercise, let I=W/H$^2$ denote the BMI described in the earlier example.

Suppose that (W, H)=(34,1.3). Use the Linear Approximation to estimate the increase in H required to keep I constant if W increases to 35.

The problem below concerns a rotating solid body and its moment of inertia about an axis; this moment relates angular acceleration to torque (an analogue of force). For a body of constant density and mass $m$ occupying a region W of volume V, the moments of inertia about the coordinate axes are

$$\begin{aligned} I_x & =\frac{m}{V} \int_W\left(y^2+z^2\right) d V \quad I_y=\frac{m}{V} \int_W\left(x^2+z^2\right) d V \\ I_z & =\frac{m}{V} \int_W\left(x^2+y^2\right) d V \end{aligned}$$

Find the moment of inertia about the z-axis of the rectangular solid of mass $m$ given by $0 \leq x \leq 1,0 \leq y \leq 2$, $0 \leq z \leq 3$

In the given exercise, let I=W/H$^2$ denote the BMI described in the earlier example.

A child has weight W=34 kg and height H=1.3 m. Use the Linear Approximation to estimate the change in I if (W, H) changes to (36,1.32).

A body of mass $M$, carrying charge $Q$, falls from rest from a height $h$ (above the ground) near the surface of the Earth, where the gravitational acceleration is $g$ and there is an electric field with a constant component $E$ in the vertical direction. a) Find an expression for the speed, $v$, of the body when it reaches the ground, in terms of $M, Q, h, g$, and $E$. b) The expression from part (a) is not meaningful for certain values of $M, g, Q$ and $E$. Explain what happens in such cases.

The system is released from rest in the configuration shown at time $t=0$. Determine the time $t$ when the block of mass $m_1$ contacts the lower stop of the body of mass $m_2$. Also, determine the corresponding distance $s_2$ traveled by $m_2$. Use the values $m_1=0.5 \mathrm{~kg}, m_2=2 \mathrm{~kg}, \mu_s=0.25, \mu_k=$ $0.20$, and $d=0.4 \mathrm{~m}$.