Angular and Linear Velocity Definitions




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Precalculus A Worksheet 09-01 Name _____________________

Angular and Linear Velocity I

Section 6.1 Date _________ Period ______
Angular and Linear Velocity
Definitions:


  • The angular velocity of a point on a rotating object is the number of degrees (radians, revolutions, etc.) per unit of time through which the point turns.




  • The linear velocity of a point on a rotating object is the distance per unit of time that the point travels along its circular path.

Note that the angular velocity of all points on a rotating object is the same. However, the linear velocity depends on how far the point is from the axis of rotation.


To see how linear velocity depends on the distance from the center, suppose the point p is r units from the center, as shown in the diagram below. Suppose, too, that the object turns through an angle of measure in t units of time. The point p, therefore, follows along the path of the arc s. Let (omega) stand for the angular velocity of p and let v stand for its linear velocity. By the definitions of angular and linear velocity:

There is simple relationship between angular and linear velocity. If is the radian measure of an angle, then the length of the corresponding arc on a unit circle is also (see the figure below). By the properties of similar geometric figures, .

Multiplying by r gives the relationship

for in radians.

Dividing each member of this equation by t gives .

Because , and , therefore,

for in radians per unit of time.







Example 1: The pedals on a bicycle turn the front sprocket at 8 radians per second. The sprocket has a diameter of 20 cm. The rear sprocket (connected to the rear bicycle wheel) has a diameter of 6 centimeters.


a) Find the linear velocity of the chain. Note that the chain is connected to the rim (outside edge) of the front sprocket. Therefore, its linear velocity is the same as the linear velocity of the rim of the sprocket.


b) Find the angular velocity of the rear sprocket. Note that the chain is also connected to the rim of the rear sprocket, which therefore must have the same linear velocity as the chain.

Note that the wheels of a bicycle rotate faster than the pedals, in this case radians per second compared to the pedal angular velocity of 8 radians per second.


Example 2: A CD is rotating at 458 revolutions per minute (rpm).

a) How many radians per minute is this?

b) Find the angular and linear velocities at a point 25mm from the center.
c) CD’s are recorded starting near the center of the disk and move towards the outside. They are recorded and read at a constant linear velocity, so the disk spins at different speeds in order to do so. As the recording track gets closer to the outside of the disk, will the disk need to speed up, or slow down?

d) Given the constant linear velocity you just calculated above, when the CD track is 50mm from the center of the disk, approximately how fast should the CD be spinning?

e) What conclusions can you draw from the above information?
Conclusions:


  • If two rotating objects are connected by their rims (as in the bicycle example) then their rims have the same linear velocity.

  • All points on the same rotating object have the same angular velocity.

  • If two rotating objects are connected so that they rotate as a single object, such as by an axle (like the bicycle rear sprocket and rear wheel), then they have the same angular velocity.


Problem Set

1. Ship’s Propeller: The propellers on an average freighter have a radius of 4 feet (see the drawing at the right). At full speed, the propellers turn at 150 revolutions per minute.


a) What is the angular velocity in radians per minute at the tip of the blades? What is it at the center of the propeller?

b) What is the linear velocity in feet per minute at the tip of the blades? What is it at the center of the propeller?

2. David and Goliath: David puts a rock in his sling and starts whirling it around. He realizes that for the rock to reach Goliath, it must leave the sling at a speed of 60 feet per second. So he swings the sling in a circular path of radius 4 feet. What must the angular velocity be in order for David to reach his objective?

3. Snowblower Cord: Yank Hardy pulls the cord on his snowblower. In order for the engine to start, the pulley must turn at 180 revolutions per minute. The pulley has a radius of 3 inches (see the diagram at the right).


a. How many radians per second must the pulley turn?

b. How fast must Yank pull the cord to start the mower? Give your answer in feet per second.


c. When Yank pulls this hard, what is the angular velocity of the center of the pulley

4. Pulley Problem: A small pulley 6 centimeters in diameter is connected by a belt to a larger pulley 15 centimeters in diameter, as show in the figure below. The small pulley is turning at 120 rpm.


a. Find the angular velocity of the small pulley in radians per second.
b. Find the linear velocity of the rim of the small pulley.

c. What is the linear velocity of the rim of the large pulley? Explain.


d. Find the angular velocity of the large pulley in radians per second.

e. How many rpm is the large pulley turning?

5. Gear Problem: A small gear of radius 5 cm is turning with an angular velocity of 20 radians per second. It drives a large gear of radius 15 cm, as shown in the diagram.


a. What is the linear velocity of the teeth on the large gear?

b. What is the angular velocity of the teeth on the large gear?


c. What is the angular velocity of a point at the center of the large gear?

6. Ant Problem: An ant is sitting 4 cm from the center of a lazy Susan (a tray that spins in a circle). Unaware of the ant, Adam spins the lazy Susan through an angle of 120 degrees.
a. Through how many radians did the ant turn?

b. What distance did the ant travel?

c. If Adam turned the lazy Susan 120 degrees in second, what was the ant’s angular velocity?

d. What was the ant’s linear velocity?

e. How do you think the ant feels about all of this spinning that’s going on? ;)

Precalculus A Worksheet 09-01 Answer Key

Angular and Linear Velocity I

Section 6.1


Angular and Linear Velocity
Definitions:

  • The angular velocity of a point on a rotating object is the number of degrees (radians, revolutions, etc.) per unit of time through which the point turns.




  • The linear velocity of a point on a rotating object is the distance per unit of time that the point travels along its circular path.

Note that the angular velocity of all points on a rotating object is the same. However, the linear velocity depends on how far the point is from the axis of rotation.


To see how linear velocity depends on the distance from the center, suppose the point p is r units from the center, as shown in the diagram below. Suppose, too, that the object turns through an angle of measure in t units of time. The point p, therefore, follows along the path of the arc s. Let (omega) stand for the angular velocity of p and let v stand for its linear velocity. By the definitions of angular and linear velocity:





There is simple relationship between angular and linear velocity. If is the radian measure of an angle, then the length of the corresponding arc on a unit circle is also (see the figure below). By the properties of similar geometric figures, .

Multiplying by r gives the relationship

for in radians.

Dividing each member of this equation by t gives .

Because , and , therefore,

for in radians per unit of time.







Example 1: The pedals on a bicycle turn the front sprocket at 8 radians per second. The sprocket has a diameter of 20 cm. The rear sprocket (connected to the rear bicycle wheel) has a diameter of 6 centimeters.


a) Find the linear velocity of the chain. Note that the chain is connected to the rim (outside edge) of the front sprocket. Therefore, its linear velocity is the same as the linear velocity of the rim of the sprocket.




b) Find the angular velocity of the rear sprocket. Note that the chain is also connected to the rim of the rear sprocket, which therefore must have the same linear velocity as the chain.

Note that the wheels of a bicycle rotate faster than the pedals, in this case radians per second compared to the pedal angular velocity of 8 radians per second.




Example 2: A CD is rotating at 458 revolutions per minute (rpm).

a) How many radians per minute is this?



b) Find the angular and linear velocities at a point 25mm from the center.






c) CD’s are recorded starting near the center of the disk and move towards the outside. They are recorded and read at a constant linear velocity, so the disk spins at different speeds in order to do so. As the recording track gets closer to the outside of the disk, will the disk need to speed up, or slow down?



d) Given the constant linear velocity you just calculated above, when the CD track is 50mm from the center of the disk, approximately how fast should the CD be spinning?

e) What conclusions can you draw from the above information?




Conclusions:


  • If two rotating objects are connected by their rims (as in the bicycle example) then their rims have the same linear velocity.

  • All points on the same rotating object have the same angular velocity.

  • If two rotating objects are connected so that they rotate as a single object, such as by an axle (like the bicycle rear sprocket and rear wheel), then they have the same angular velocity.


Problem Set

1. Ship’s Propeller: The propellers on an average freighter have a radius of 4 feet (see the drawing at the right). At full speed, the propellers turn at 150 revolutions per minute.


a) What is the angular velocity in radians per minute at the tip of the blades? What is it at the center of the propeller?


b) What is the linear velocity in feet per minute at the tip of the blades? What is it at the center of the propeller?

2. David and Goliath: David puts a rock in his sling and starts whirling it around. He realizes that for the rock to reach Goliath, it must leave the sling at a speed of 60 feet per second. So he swings the sling in a circular path of radius 4 feet. What must the angular velocity be in order for David to reach his objective?




3. Snowblower Cord: Yank Hardy pulls the cord on his snowblower. In order for the engine to start, the pulley must turn at 180 revolutions per minute. The pulley has a radius of 3 inches (see the diagram at the right).
a. How many radians per second must the pulley turn?


b. How fast must Yank pull the cord to start the mower? Give your answer in feet per second.
c. When Yank pulls this hard, what is the angular velocity of the center of the pulley

4. Pulley Problem: A small pulley 6 centimeters in diameter is connected by a belt to a larger pulley 15 centimeters in diameter, as show in the figure below. The small pulley is turning at 120 rpm.


a. Find the angular velocity of the small pulley in radians per second.
b. Find the linear velocity of the rim of the small pulley.

c. What is the linear velocity of the rim of the large pulley? Explain.




d. Find the angular velocity of the large pulley in radians per second.



e. How many rpm is the large pulley turning?

5. Gear Problem: A small gear of radius 5 cm is turning with an angular velocity of 20 radians per second. It drives a large gear of radius 15 cm, as shown in the diagram.
a. What is the linear velocity of the teeth on the large gear?


b. What is the angular velocity of the teeth on the large gear?


c. What is the angular velocity of a point at the center of the large gear?

6. Ant Problem: An ant is sitting 4 cm from the center of a lazy Susan (a tray that spins in a circle). Unaware of the ant, Adam spins the lazy Susan through an angle of 120 degrees.


a. Through how many radians did the ant turn?


b. What distance did the ant travel?

c. If Adam turned the lazy Susan 120 degrees in second, what was the ant’s angular velocity?


d. What was the ant’s linear velocity?



e. How do you think the ant feels about all of this spinning that’s going on? ;)





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