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### Chapter 7

Rotational Motion

Measuring 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

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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:

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- To convert from degrees to radians:

Measuring Rotational Motion

- 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

Measuring Rotational Motion

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

Measuring Rotational Motion

- 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?

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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.

Measuring Rotational Motion

- 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.

Measuring Rotational Motion

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

Measuring Rotational Motion

- 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?

Measuring Rotational Motion

- 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?

Measuring Rotational Motion

- 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?

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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

Measuring Rotational Motion

- 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?

Tangential and Centripetal Acceleration

- 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 and Centripetal Acceleration

- 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

Tangential and Centripetal Acceleration

- 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

Tangential and Centripetal Acceleration

- 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 and Centripetal Acceleration

- 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:

Tangential and Centripetal 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?

Tangential and Centripetal Acceleration

- 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

Tangential and Centripetal Acceleration

- To find the centripetal acceleration:

Tangential and Centripetal Acceleration

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

Tangential and Centripetal Acceleration

- 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 Acceleration

- 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

Causes of Circular Motion

- 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

Causes of Circular Motion

- Force that maintains circular motion can be found by:

Causes of Circular Motion

- 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

Causes of Circular Motion

- 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.

Causes of Circular Motion

- 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

Causes of Circular Motion

- 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.

Causes of Circular Motion

- 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

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