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# Chapter 7 - PowerPoint PPT Presentation

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

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## PowerPoint Slideshow about ' Chapter 7' - dorinda-urbina

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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 2rad

• 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 2r

• 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 2rad

• 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