Tangential and Centripetal Acceleration

TANGENTIAL SPEED

TANGENTIAL SPEEDvt= rω

tangential speed = distance from axis x angular speed

Note that ω is the instantaneous angular speed, rather than the average angular speed, because the time interval is so short. This equation is valid only when co is measured in radians per unit of time. Other measures of angular speed, such as degrees per second and revolutions per second, must not be used in this equation.

Tangential speedPROBLEMThe radius of a CD in a computer is 0.0600 m. If a microbe riding on the

disc's rim has a tangential speed of 1.88 m/s, what is the disc's angular speed?

SOLUTIONGiven: r= 0.0600 m vt = 1.88 m/s

Unknown: ω =?

Use the tangential speed equation on this page to solve for angular speed.vt= rω

Problems

1. A woman passes through a revolving door with a tangential speed of 1.8 m/s. If she is 0.80 rn from the center of the door, what is the door's angular speed?

2. An athlete spins in a circle before releasing a discus with a tangential speed of 9.0 m/s. What is the angular speed of the spinning athlete? Assume the discus is 0.75 m from the athlete's axis of rotation.

Practice/Assignment

4. A cylinder with a diameter of 0.457 m rotates in a lathe at a constant angular speed of 35 rad/s. What is the tangential speed of the surface of the cylinder?

5. A wheel with a radius of 2.0 m rotates at a constant angular speed of 15.5 rad/s. What is the tangential speed of a point 0.55 m from the wheel's axis?

Assignment

vt ω r

a ? 121.5 rad/s 0.030 m

b 0.75 m/s ? 0.050 m

c ? 1.2 turns/s 3.8 m

d 2.0π m/s 1.5π rad/s ?

2. A softball pitcher throws a ball with a tangential speed of 6.93 m/s. If the pitcher's arm is 0.660 rn long, what is the angular speed of the ball before the pitcher releases it?

4. Fill in the unknown quantities in the following table:

Tangential Acceleration

If the time interval t is very small, then the left side of this relationship gives the tangential acceleration of the point.

The angular speed divided by the time interval on the right side is the angular acceleration.

TANGENTIAL ACCELERATION at = rα

tangential acceleration = distance from axis x angular acceleration

Again, the angular acceleration in this equation refers to the instantaneous angular acceleration. This equation must use the unit radians to be valid. In SI, angular acceleration is expressed as radians per second per second.

Tangential acceleration

PROBLEMA spinning ride at a carnival has an angular acceleration of 0.50 rad/s2. How far from the center is a rider who has a tangential acceleration of 3.3 m/s2? SOLUTIONGiven: α = 0.50 rad/s2 at=3.3 m/s

Unknown: r=?

Use the tangential acceleration equation on this page. Rearrange to solve for r.

at= rα

Practice problems

1. A dog on a merry-go-round undergoes a 1.5 m/s2 linear acceleration. If the merry-go-round's angular acceleration is 1.0 rad/s2 , how far is the dog from the axis of rotation?

2. A young boy swings a yo-yo horizontally above his head at an angular acceleration of 0.35 rad/s2. If tangential acceleration of the yo-yo at the end of the string is 0.18 m/s2, how long is the string?

3. What is a tire's angular acceleration if the tangential acceleration at a radius of 0.15 m is 9.4 X 10-2 m/s2?

Centripetal Acceleration

PROBLEA test car moves at a constant speed around a circular

track. If the car is 48.2 m from the track's center and has a centripetal acceleration of 8.05 m/s2, what is its tangential speed?

SOLUTIONGiven: r= 48.2m ac= 8.05 m/s2

Unknown: vt

Practice problems

1. A girl sits on a tire that is attached to an overhanging tree limb by a rope. The girl's father pushes her so that her centripetal acceleration is 3.0 m/s . If the length of the rope is 2.1 m, what is the girl's tangential speed?

2. A dog sits 1.5 m from the center of a merry-go-round. If the dog undergoes a 1.5 m/s centripetal acceleration, what is the dog's linear speed? What is the angular speed of the merry-go-round?

3. A piece of clay sits 0.20 m from the center of a potter's wheel. If the potter spins the wheel at an angular speed of 20.5 rad/s, what is the magnitude of the centripetal acceleration of the piece of clay on the wheel?