A basketball player is balancing a spinning basketball on the tip of his finger. The angular velocity of the ball slows down from $18.5$ to $14.1 \mathrm{rad} / \mathrm{s}$. During the slow-down, the angular displacement is $85.1 \mathrm{rad}$. Determine the time it takes for the ball to slow down.

You are told that a basketball player spins the ball with an angular acceleration of 100 rad/s$^2$. (a) What is the ball's final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke's law and is depressed 0.75 cm by its maximum load of 120 kg. (a) What is the spring's effective force constant? (b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85-kg-team?

A string is wrapped around a wheel of radius 25 cm mounted on a stationary axle. The wheel is initially not rotating. You pull the string with a constant force through a distance of 40 cm. What is the angle in radians and degrees through which the wheel rotates?

An ice cube floats in a glass of water. When the ice melts, will the water level in the glass rise, fall, or remain unchanged? Explain.

A 220-kg speedboat is negotiating a circular turn (radius = 32 m) around a buoy. During the turn, the engine causes a net tangential force of magnitude 550 N to be applied to the boat. The initial tangential speed of the boat going into the turn is 5.0 m/s. (a) Find the tangential acceleration. (b) After the boat is 2.0 s into the turn, find the centripetal acceleration.

Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is

$$2.2 \times 10 ^ { 20 }$$

and the angular speed of the sun is

$$1.2 \times 10 ^ { - 15 } \mathrm { rad } / \mathrm { s }$$

(a) What is the tangential speed of the sun? (b) How long (in years) does it take for the sun to make one revolution around the center?

A ball of radius 0.200 m rolls along a horizontal table top with a constant linear speed of 3.60 m/s. The ball rolls off the edge and falls a vertical distance of 2.10 m before hitting the floor. What is the angular displacement of the ball while the ball is in the air?

Over the course of a multi-stage 4520-km bicycle race , the front wheel of an athlete's bicycle makes $2.18 \times 10^{6}$ revolutions. How many revolutions would the wheel have made during the race if its radius had been 1.2 cm larger?

The two-gear combination shown in the drawing is being used to lift the load $L$ with a constant upward speed of $2.50 \mathrm{~m} / \mathrm{s}$. The rope that is attached to the load is being wound onto a cylinder behind the big gear. The depth of the teeth of the gears is negligible compared to the radii. Determine the angular velocity (magnitude and direction) of (a) the larger gear and (b) the smaller gear.

Two identical dragsters, starting from rest, accelerate side-by-side along a straight track. The wheels on one of the cars roll without slipping, while the wheels on the other slip during part of the time. (a) For which car, the winner or the loser, do the wheels roll without slipping? Why? For the dragster whose wheels roll without slipping, is there (b) a relationship between its linear speed and the angular speed of its wheels, and (c) a relationship between the magnitude of its linear acceleration and the magnitude of the angular acceleration of its wheels? If a relationship exists in either case, what is it? A dragster starts from rest and accelerates down the track. Each tire has a radius of 0.320 m and rolls without slipping. At a distance of 384 m, the angular speed of the wheels is 288 rad/s. Determine (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.

A motorcycle, which has an initial linear speed of 6.6 m/s, decelerates to a speed of 2.1 m/s in 5.0 s. Each wheel has a radius of 0.65 m and is rotating in a counterclockwise (positive) direction. What is (a) the constant angular acceleration

$$\left( \text { in } \mathrm { rad } / \mathrm { s } ^ { 2 } \right)$$

and (b) the angular displacement (in rad) of each wheel?

An automobile tire has a radius of 0.330 m, and its center moves forward with a linear speed of

$$v = 1.50m/s$$

(a) Determine the angular speed of the wheel. (b) Relative to the axle, what is the tangential speed of a point located 0.175 m from the axle?

A car is traveling with a speed of 20.0 m/s along a straight horizontal road. The wheels have a radius of 0.300 m. If the car speeds up with a linear acceleration of

$$1.50 m/s^2$$

for 8.00 s, find the angular displacement of each wheel during this period.