A square metal plate 0.180 m on each side is pivoted about an axis through point O at its center and perpendicular to the plate. Calculate the net torque about this axis due to the three forces shown in the figure if the magnitudes of the forces are $\text{F}_{ 1 } = 18.0 \ \text{N}$, $\text{F}_{ 2 } = 26.0 \text{ N }$, and $\text{F}_3 = 14.0 \ \text{N}$. The plate and all forces are in the plane of the page.

Calculate the net torque about point $O$ for the two forces applied as in Fig. The rod and both forces are in the plane of the page.

Three forces are applied to a wheel of radius 0.350, as shown in the figure (Intro 1figure). One force is perpendicular to the rim, one is tangent to it, and the other one makes a 40.0 angle with the radius. Assume that $F_1=12.4~\text{N}$, $F_2=14.2~\text{N}$, and $F_3=8.75~\text{N}$.

Part A What is the magnitude of the net torque on the wheel due to these three forces for an axis perpendicular to the wheel and passing through its center?

Part B What is the direction of the net torque in part (A), into the page or out of the page?

A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass 0.180 kg. The free end of the string is held in place and the hoop is released from rest. After the hoop has descended 75.0 cm, calculate the speed of its center.

One force acting on a machine part is $\vec{F}=(-5.00 \mathrm{~N}) \hat{\imath}+$ $(4.00 \mathrm{~N}) \hat{\jmath}$. The vector from the origin to the point where the force is applied is $\vec{r}=(-0.450 \mathrm{~m}) \hat{\imath}+(0.150 \mathrm{~m}) \hat{\jmath}$. (a) In a sketch, show $\vec{r}$, $\vec{F}$, and the origin.

A bowling ball rolls with-out slipping up a ramp that slopes upward at an angle $\beta$ to the horizontal. Treat the ball as a uniform solid sphere, ignoring the finger holes. (a) Draw the free-body diagram for the ball. Explain why the friction force must be directed uphill. (b) What is the acceleration of the center of mass of the ball? (c) What minimum coefficient of static friction is needed to prevent slipping?

Approximately 11% of each batch of yo-yos is defective. Mr. Andersen said that in a batch of 1500 yo-yos, 150 yo-yos would be defective. Estimate to determine if Mr. Andersen's number is reasonable. Explain.

Calculate the magnitude of the angular momentum of the earth due to its rotation around an axis through the north and south poles, modeling it as a uniform sphere.

A 60.0-cm, uniform, 50.0-N shelf is supported horizontally by two vertical wires attached to the sloping ceiling. A very small 25.0-N tool is placed on the shelf midway between the points where the wires are attached to it. Find the tension in each wire. Begin by making a free-body diagram of the shelf.

True or False? Justify your answer with a proof or a counterexample. You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

A square metal plate of edge length 8.0 cm and negligible thickness has a total charge of $6.0 \times 10^{-6} C$. Estimate the magnitude E of the electric field just off the center of the plate (at, say, a distance of 0.50 mm from the center) by assuming that the charge is spread uniformly over the two faces of the plate.

A solid, uniform, spherical boulder starts from rest and rolls down a 50.0-m-high hill. The top half of the hill is rough enough to cause the boulder to roll without slipping, but the lower half is covered with ice and there is no friction. What is the translational speed of the boulder when it reaches the bottom of the hill?

A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of 0.0200 N at that point. (a) Find its angular acceleration. (b) How long will it take to decrease its rotational speed by 22.5 rad/s?

A classic $1957$ Chevrolet Corvette of mass $1240 \mathrm{~kg}$ starts from rest and speeds up with a constant tangential acceleration of $2.00 \mathrm{~m} / \mathrm{s}^2$ on a circular test track of radius $60.0 \mathrm{~m}$. Treat the car as a particle. $(a)$ What is its angular acceleration? $(b)$ What is its angular speed $6.00 \mathrm{~s}$ after it starts? $(c)$ What is its radial acceleration at this time? $(d)$ Sketch a view from above showing the circular track, the car, the velocity vector, and the acceleration component vectors $6.00 \mathrm{~s}$ after the car starts. $(e)$ What are the magnitudes of the total acceleration and net force for the car at this time? $(f)$ What angle do the total acceleration and net force make with the car's velocity at this time?