# Centripetal acceleration, Angular speed and Tangential speed

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1. When jumping straight down, you can be seriously injured if you land stiff-legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A 80.4-kg man just before contact with the ground has a speed of 8.27 m/s.

(a) In a stiff-legged landing he comes to a halt in 3.85 ms. Find the magnitude of the average net force that acts on him during this time.

(b) When he bends his knees, he comes to a halt in 0.194 s. Find the magnitude of the average net force now.

(c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the magnitude of the force applied by the ground on the man in part (b).

2. A car travels at a constant speed around a circular track whose radius is 2.49 km. The car goes once around the track in 212 s. What is the magnitude of the centripetal acceleration of the car?

3. The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located 19.0 m from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 5.39 times the acceleration due to gravity?

4. In a skating stunt known as "crack-the-whip", a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of 76.0 kg and is 6.60 m from the pivot. He is skating at a speed of 5.00 m/s. Determine the magnitude of the centripetal force that acts on him.

5. At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of 2.94 m/s, and an 82.2-kg person feels a 380-N force pressing against his back. What is the radius of a chamber?

6. A pitcher throws a curve-ball that reaches the catcher in 0.54 s. The ball curves because it is spinning at an average angular velocity of 280 rev/min (assumed constant) on its way to the catcher's mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

7. A Ferris wheel rotates at an angular velocity of 0.22 rad/s. Starting from rest, it reaches its operating speed with an average angular acceleration of 0.029 rad/s^2. How long does it take the wheel to come up to operating speed?

8. The drill bit of a variable-speed electric drill has a constant angular acceleration of 3.41 rad/s^2. The initial angular speed of the bit is 6.07 rad/s. After 4.59 s, (a) what angle has the bit turned through and (b) what is the bit's angular speed?

9. A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon "string" that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 48 rev/s, and its tip has a tangential speed of 66 m/s. What is the length of the rotating string?

10. An auto race is held on a circular track. A car completes one lap in a time of 24.9 s, with an average tangential speed of 40.6 m/s. Find (a) the average angular speed and (b) the radius of the track.

11. The take-up reel of a cassette tape has an average radius of 1.5 cm. Find the length (in m) of tape that passes around the reel in 16 s when the reel rotates at an average angular speed of 2.1 rad/s.

12. Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 10.5 rad/s. The wheel has a radius of 0.401 m. If you ride the bike for 47.7 min, how far would you have gone if the bike could move?

13. An automobile tire has a radius of 0.328 m, and its center moves forward with a linear speed of v = 20.8 m/s.

(a) Determine the angular speed of the wheel.

(b) Relative to the axle, what is the tangential speed of a point located 0.248 m from the axle?

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Centripetal acceleration, angular speed and tangential speed is examined.

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