With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s under a constant angular acceleration. (a) What is its angular acceleration in rad/s²? (b) How many revolutions does it go through in the process?

A cruise ship with a mass of

$$1.00 × 10^7 kg$$

strikes a pier at a speed of 0.750 m/s. It comes to rest after traveling 6.00 m, damaging the ship, the pier, and the tugboat captain's finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest, assuming a constant force.)

At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

Suppose a piece of dust has fallen on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)

Suppose you start an antique car by exerting a force of 300 N on its crank for 0.250 s. What is the angular momentum given to the engine if the handle of the crank is 0.300 m from the pivot and the force is exerted to create maximum torque the entire time?

Describe two factors that cause real gases to depart from the ideal gas law.

Evaluate the integrals Sketch and identify the type of the region (corresponding to the way the integral is written).

$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x ^ { 2 } } \left( x ^ { 2 } + x y - y ^ { 2 } \right) d y d x$$

Compute (a) the general solution and (b) the solution with $y(0)=y^{\prime}(0)=0$.

$$\frac{d^2 y}{d t^2}+4 \frac{d y}{d t}+6 y=-8$$

Show that

$$\frac{d t}{d x}=\frac{x}{c_{1} \sqrt{a^{2}+x^{2}}}-\frac{d-x}{c_{2} \sqrt{b^{2}+(d-x)^{2}}}$$

is an increasing function of x.

One cart of mass $12.0 \mathrm{~kg}$ is moving $6.00 \mathrm{~m} / \mathrm{s}$ to the right on a frictionless track and collides with a cart of mass $4.00 \mathrm{~kg}$ moving in the opposite direction $3.00 \mathrm{~m} / \mathrm{s}$. What is the final velocity of the carts that become stuck together after the collision.

According to Mars, Incorporated, 20% of its plain M&M'S candies are orange. Assume that the company's claim is true. Suppose that you reach into a large bag of plain M&M'S (without looking) and pull out 8 candies. Let X = the number of orange candies you get. Would you be surprised if none of the candies were orange? Compute an appropriate probability to support your answer.

In this exercise, find the points of intersection of the graphs of the equations.

$$x^2+y^2=25$$

$$2x+y=10$$

The 150 -lb skater passes point $A$ $(y = 20 \text{ ft})$ with a speed of $6 \mathrm{ft} / \mathrm{s}$. Determine his speed when he reaches point $B$ $( x= 25 \text{ ft}$) and the normal force exerted on him by the track at this point. The path $AB$ is defined by the equation $y^2 = 4x$. Neglect friction.

If $\theta=1 \mathrm{rad}, d \theta / d t=-2 \mathrm{rad} / \mathrm{s}$, and $d^2 \theta / d t^2=0$, what are the velocity and acceleration of $P$ relative to $O$ ?