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Chapter 7

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Chapter 7

Rotational Motion

- When an object spins it is said to undergo rotational motion.
- The axis of rotation is the line about which the rotation occurs
- A point on an object that rotates about a single axis undergoes circular motion around that axis
- Any given point on an object of any shape rotates about the axis in a circular pattern

- Difficult to describe the motion of an object in rotational motion using only information for the linear case
- This is because the path is consistently changing

- When rotational motion is described using angles, all points on a rigid body move through the same angle in the same interval of time

- To analyze rotational motion, choose a fixed reference line
- The distance a point moves is called the arc length and is denoted by the variable s

s

r

Reference Line

r

Reference Line

- Angles can be measured in radians
- Prior to this point, we have used degrees
- A radianis an angle whose arc length is equal to its radius, which is approximately equal to 57.3
- Be sure to convert to radians in this chapter!!
- Defined as an equation:

- Because radius is defined as arc length divided by radius, both are distances
- The units are canceled and the abbreviation rad is used in their place
- 360 is equal to 2rad
- The arc length s is equal to the circumference of a circle, if the object travels the full rotation
- The circumference of a circle is 2r

- To convert from degrees to radians:

- Angular displacement describes how much an object has rotated
- The angular displacement traveled is equal to the change in arc length divided by the radius from the axis of rotation to that point

- In general, a positive arc length is taken to be rotating counter clockwise and a negative arc length rotates clockwise

- Practice Problem 1
- Earth has an equatorial radius of approximately 6380 km and rotates 360 in 24h.
- What is the angular displacement, in degrees, of a person standing on the equator for 1.0h?
- Covert this angular displacement to radians.
- What is the arc length traveled by this person?

- Angular speed describes the rate of rotation
- The average angular speed of a rotating rigid object is the ration of the angular displacement to the time interval it takes for the object to undergo that displacement
- It describes how quickly the rotation occurs

- Angular speed is given in units of radians per second (rad/s)
- Sometimes given in revolutions per unit time
- Remember that 1 revolution is 2rad

- Practice Problem 2
- Before the advent of compact discs, musical recordings were commonly sold on vinyl discs that could be played on a turntable at 45 rpm (revolution per minute) or 33.3 rpm. Calculate the corresponding angular speeds in rad/s.

- Practice Problem 3
- An Indy car can complete 120 laps in 1.5h. Even though the track is oval rather than a circle, you can still find the average angular speed. Calculate the average angular speed of the Indy car in rad/s.

- Angular acceleration occurs when angular speed changes
- The average angular acceleration is given below.
- The units for angular acceleration are rad/s2

- Practice Problem 4
- A yo-yo at rest is sent spinning at an angular speed of 12 rev/s in 0.25 s. What is the average angular acceleration of the yo-yo?

- Practice Problem 5
- A top spinning at 15 rev/s spins for 55 s before coming to a stop. What is the average angular acceleration of the top while it is slowing?

- Practice Problem 6
- A certain top will remain stable (upright) at angular speeds above 3.5 rev/s. The top slows due to friction at a rate of 1.3 rad/s2. What initial speed (in rad/s and rev/s) must the top be given in order to spin for at least 1 min?

- All points on a rotating rigid object have the same angular acceleration and angular speed
- For a rotating object to remain rigid every portion of the object must have the same angular speed and the same angular acceleration, if not then the object changes shape

- Compare the angular motion equations to those we already know for linear motion
- The equations are similar, however the rotational equations replace linear variables with their rotational counterparts

- Practice Problem 7
- A barrel is given a downhill rolling start of 1.5 rad/s at the top of a hill. Assume a constant angular acceleration of 2.9 rad/s2.
- If it takes 11.5 s to get to the bottom of the hill, what is the final angular speed of the barrel?
- What angular displacement does the barrel experience during the 11.5 s ride?

- Sometimes it is useful to describe the motion of a rotating object in terms of linear speed and linear acceleration of a single point on that object
- Objects in circular motion have a tangential speed
- Tangential speed is the instantaneous linear speed of an object directed along the tangent to the objects circular path

- Tangential speed is also known as the instantaneous linear speed of that point
- The tangential speed of two different points at different distances from the center have different tangential speeds
- If the angular displacement is the same and the radius increases, the arc length traveled must also increase. Therefore the point on the outside has a larger tangential speed

- To find tangential speed:
- Note that is the instantaneous angular speed, not the average angular speed
- This is only valid when is measured in radian

- Practice Problem 8
- A golfer has a maximum angular speed of 6.3 rad/s for her swing. She can choose between two drivers, one placing the club head 1.9 m from her axis or rotation and the other placing it at 1.7 m from the axis
- Find the tangential speed of the club head for each driver.
- All other factors being equal, which driver is likely to hit the ball farther?

- Tangential acceleration is tangent to the circular path
- The tangential acceleration is the instantaneous linear acceleration of an object directed along the tangent to the objects circular path
- To find the tangential acceleration:

- Practice Problem 9
- A yo-yo has a tangential acceleration of 0.98 m/s2 when it is released. The string is wound around the central shaft of radius 0.35 cm. What is the angular acceleration of the yo-yo?

- If you are moving around a circle with a constant tangential speed, you are still accelerating
- This is because you are changing direction
- The acceleration caused by the change in direction is called centripetal acceleration
- Centripetal acceleration is an acceleration directed toward the center of a circular path

- To find the centripetal acceleration:

- DO NOT replace centripetal with centrifugal, centripetal mean center seeking and centrifugal means center fleeing. They are opposite in meaning!!!

- Practice Problem 10
- A cylindrical space station with a 115 m radius rotates around its longitudinal axis at an angular speed of 0.292 rad/s. Calculate the centripetal acceleration on a person at the following locations
- At the center of the station
- Halfway to the rim of the station
- At the rim of the station

- Tangential and Centripetal accelerations are perpendicular to each other
- Tangential acceleration is due to changing speed
- Centripetal acceleration is due to changing direction
- You may find the magnitude of the total acceleration by using Pythagoreans theorem
- You can also find the direction by using the inverse tangent function

- The inertia of an object tends to maintain the object’s motion in a straight line path
- Circular motion is possible because of the force that is directed towards the axis of rotation
- This force can be found by applying Newton’s Second Law in the radial direction

- Force that maintains circular motion can be found by:

- The force needed to maintain circular motion is no different than any other force we have discussed
- An example of this can be seen where the tires of a car encounter friction in order to make it move in a circular path

- Practice Problem 11
- An astronaut who weighs 735 N on Earth is at the rim of a cylindrical space station with a 73 m radius. The space station is rotating at an angular speed of 3.5 rpm. Evaluate the force that maintains the circular motion of the astronaut.

- A force directed toward the center is necessary for circular motion
- If this force vanishes, the object does not continue to move in a circular path, but it continues in a straight line path that is tangent to the circular path it was in

- Once this force vanishes, it continues in motion as we have studied previously
- For example, if a ball was attached to a string being swung vertically in a circle and the string broke at the top of its path, the ball would continue as a projectile launched horizontally and you can continue to solve the problem as before.

- Describe what happens when a car makes a fast turn.
- What causes the passenger to move toward the door?
- INERTIA
- The passenger is originally moving in a straight line path
- When the car makes the turn, the passenger wants to continue in the straight line path until an outside force create a change in the direction of the person
- The force of the door on the person is what makes the person turn and follow the circular path

- What causes the passenger to move toward the door?