Circular MotionCircular Motion

Unit 5Unit 5

An axis is the straight line around An axis is the straight line around which rotation takes place. When which rotation takes place. When an object turns about an internal an object turns about an internal

axis- that is, an axis located within axis- that is, an axis located within the body of the object- the motion is the body of the object- the motion is

called rotation.called rotation.When an object turns about an When an object turns about an

external axis, the motion is called external axis, the motion is called revolution.revolution.

Earth undergoes both types of Earth undergoes both types of rotational motion. It revolves rotational motion. It revolves around the sun once every around the sun once every 365 ¼ days and it rotates 365 ¼ days and it rotates

around an axis passing through around an axis passing through its geographical poles once its geographical poles once

every 24 hours.every 24 hours.

Linear speedLinear speed

Linear speed is the distance moved per unit Linear speed is the distance moved per unit of time. A point on the outer edge of a of time. A point on the outer edge of a merry-go-round moves a greater distance merry-go-round moves a greater distance in one complete rotation that a point near in one complete rotation that a point near the center. The linear speed is greater on the center. The linear speed is greater on the outer edge of rotating objects than it is the outer edge of rotating objects than it is closer to its axis. The speed of something closer to its axis. The speed of something moving along a circular path can be called moving along a circular path can be called tangential speed because the direction of tangential speed because the direction of motion is always tangent to the circle.motion is always tangent to the circle.

Rotational speedRotational speed

Rotational speed is the number of rotations Rotational speed is the number of rotations per unit time. All parts of the merry-go-per unit time. All parts of the merry-go-round rotate about their axis in the same round rotate about their axis in the same amount of time. Thus, all parts have the amount of time. Thus, all parts have the same rate of rotation. Rotational speed is same rate of rotation. Rotational speed is expressed in revolutions per minute expressed in revolutions per minute (RPM).(RPM).

Tangential speed and rotational speed are related. Tangential speed and rotational speed are related. The faster it turns, the faster your tangential speed The faster it turns, the faster your tangential speed

is. is.

Tangential speed is directly proportional to Tangential speed is directly proportional to rotational speed and the radial distance from the rotational speed and the radial distance from the

axis of rotation. At the center of a rotating platform, axis of rotation. At the center of a rotating platform, right at its axis, you have no tangential speed at right at its axis, you have no tangential speed at

all, but you do have rotational speed. As you move all, but you do have rotational speed. As you move away from the center, you move faster and faster- away from the center, you move faster and faster-

your tangential speed increases while your your tangential speed increases while your rotational speed stays the same.rotational speed stays the same.

Centripetal forceCentripetal force

Any force that causes an object to follow a Any force that causes an object to follow a circular pathcircular path

Means “center seeking” or “towards the center”Means “center seeking” or “towards the center” When an automobile rounds a corner, the When an automobile rounds a corner, the

sideways-acting friction between the tires and sideways-acting friction between the tires and the road provides the centripetal force that holds the road provides the centripetal force that holds the car on a curved path. If the force of friction is the car on a curved path. If the force of friction is not great enough, the car fails to make the curve not great enough, the car fails to make the curve and the tires slide sideways. The car skids.and the tires slide sideways. The car skids.

Centrifugal forceCentrifugal force

Outward forceOutward force Means “center fleeing” or “away from the Means “center fleeing” or “away from the

center”center”

Suppose you are whirling a can. If the string on the whirling Suppose you are whirling a can. If the string on the whirling can breaks, it if often wrongly stated that centrifugal force can breaks, it if often wrongly stated that centrifugal force pulls the can from its circular path. But in fact, when the pulls the can from its circular path. But in fact, when the string breaks the can goes off in a tangential straight-line string breaks the can goes off in a tangential straight-line

path because no force acts on it.path because no force acts on it.

Now suppose there is a ladybug inside the whirling can. Now suppose there is a ladybug inside the whirling can. The can provides the centripetal force, not the centrifugal The can provides the centripetal force, not the centrifugal force necessary to hold the ladybug in a circular path. The force necessary to hold the ladybug in a circular path. The “centrifugal-force effect” is attributed not to any real force “centrifugal-force effect” is attributed not to any real force but to inertia- the tendency of the moving body to follow a but to inertia- the tendency of the moving body to follow a

straight-line pathstraight-line path

Our view of nature depends on the frame of reference from Our view of nature depends on the frame of reference from which we view it. For instance, when sitting on a fast-which we view it. For instance, when sitting on a fast-

moving train, we have no speed at all relative to the train, moving train, we have no speed at all relative to the train, but we have an appreciable speed relative to the reference but we have an appreciable speed relative to the reference

frame of the ground outside.frame of the ground outside.

From a stationary frame outside the whirling can, we see From a stationary frame outside the whirling can, we see there is no centrifugal force acting on the ladybug. We do there is no centrifugal force acting on the ladybug. We do

see centripetal force acting on the can. In the rotating see centripetal force acting on the can. In the rotating frame of reference of the whirling can, both centripetal frame of reference of the whirling can, both centripetal

force and centrifugal force act on the ladybug. However, force and centrifugal force act on the ladybug. However, centrifugal force is an effect of rotation. It is not part of an centrifugal force is an effect of rotation. It is not part of an

interaction so it cannot be a true force.interaction so it cannot be a true force.

Gravity is simulated by centrifugal force. If Gravity is simulated by centrifugal force. If the spinning can freely falls, the ladybug the spinning can freely falls, the ladybug

inside will experience a centrifugal force that inside will experience a centrifugal force that feels like gravity when the can spins at an feels like gravity when the can spins at an

appropriate rate.appropriate rate.

Today we live on the outer surface of a spherical planet, Today we live on the outer surface of a spherical planet, held here by gravity. In the years ahead many people will held here by gravity. In the years ahead many people will

likely live in huge, lazily rotating space stations where likely live in huge, lazily rotating space stations where centrifugal force simulates gravity. The simulated gravity centrifugal force simulates gravity. The simulated gravity

will be provided so the people can function normally. will be provided so the people can function normally. Occupants in today’s space shuttles feel weightless Occupants in today’s space shuttles feel weightless

because they lack a support force. They’re not pressed because they lack a support force. They’re not pressed against a supporting floor by gravity, nor do they against a supporting floor by gravity, nor do they

experience a centrifugal force due to spinning. But future experience a centrifugal force due to spinning. But future space travelers need not be subject to weightlessness. space travelers need not be subject to weightlessness.

Their space habitats will likely spin effectively supplying a Their space habitats will likely spin effectively supplying a support force and nicely simulating gravity.support force and nicely simulating gravity.

The comfortable 1g we The comfortable 1g we experience at Earth’s surface is experience at Earth’s surface is due to gravity. Inside a rotating due to gravity. Inside a rotating

spaceship the acceleration spaceship the acceleration experienced is the centripetal/ experienced is the centripetal/ centrifugal acceleration due to centrifugal acceleration due to

rotation.rotation.

FormulasFormulas

Period: the time it takes for one full rotation or Period: the time it takes for one full rotation or revolution of an objectrevolution of an object Unit is secondsUnit is seconds

Frequency: the number of rotations or Frequency: the number of rotations or revolutions per unit timerevolutions per unit time Unit is HertzUnit is Hertz

T= T= 11

ff

FormulasFormulas

When an object spins in a circle, the When an object spins in a circle, the distance it travels in one revolution is the distance it travels in one revolution is the circumference of a circle. The time it takes circumference of a circle. The time it takes is the period.is the period.

speed= speed= 22ΠΠrr

TT

formulasformulas

An object can move around in a circle with a An object can move around in a circle with a constant speed yet still be accelerating constant speed yet still be accelerating because its direction is constantly because its direction is constantly changing. This is centripetal acceleration.changing. This is centripetal acceleration.

centripetal acc= centripetal acc= (linear speed)(linear speed)22

radiusradius

Formulas Formulas

If the mass is being accelerated towards the If the mass is being accelerated towards the center of a circle, it must be acted upon by center of a circle, it must be acted upon by an unbalanced force that gives it this an unbalanced force that gives it this acceleration.acceleration.

centripetal force= centripetal force= mvmv22

rr

After closing a deal with a client, After closing a deal with a client, Kent leans back in his swivel Kent leans back in his swivel chair and spins around with a chair and spins around with a frequency of 0.5Hz. What is frequency of 0.5Hz. What is

Kent’s period of spin?Kent’s period of spin?

Curtis’ favorite disco record has Curtis’ favorite disco record has a scratch 12cm from the center a scratch 12cm from the center that makes the record skip 45 that makes the record skip 45

times each minute. What is the times each minute. What is the linear speed of the scratch as it linear speed of the scratch as it

turns?turns?

Missy’s favorite ride at the fair is Missy’s favorite ride at the fair is the rotor, which has a radius of 4m. the rotor, which has a radius of 4m. The ride takes 2s to make one full The ride takes 2s to make one full

revolution.revolution.

What is Missy’s linear speed on the What is Missy’s linear speed on the rotor?rotor?

What is Missy’s centripetal What is Missy’s centripetal acceleration on the rotor?acceleration on the rotor?

Captain Chip, the pilot of a Captain Chip, the pilot of a 60500kg jet plane, is told that he 60500kg jet plane, is told that he must remain in a holding pattern must remain in a holding pattern

over the airport until it is his turn to over the airport until it is his turn to land. If Captain Chip flies his plane land. If Captain Chip flies his plane

in a circle whose radius is 50km in a circle whose radius is 50km once every 30 min, what centripetal once every 30 min, what centripetal force must the air exert against the force must the air exert against the wings to keep the plane moving in wings to keep the plane moving in

a circle?a circle?

FormulaFormula

Torque: a measurement of the tendency of Torque: a measurement of the tendency of a force to produce a rotation about an axisa force to produce a rotation about an axis torque= perpendicular force x lever armtorque= perpendicular force x lever arm

The lever arm is the distance from the pivot point, or The lever arm is the distance from the pivot point, or fulcrum, to the point where the component of force fulcrum, to the point where the component of force perpendicular to the lever arm is being exerted. The perpendicular to the lever arm is being exerted. The longer the lever arm, the greater the torque.longer the lever arm, the greater the torque.

Keep in mind that when an object is balanced, all torques Keep in mind that when an object is balanced, all torques must also balance.must also balance.

Ned tightens a bolt in his car Ned tightens a bolt in his car engine by exerting 12N of force engine by exerting 12N of force on his wrench at a distance of on his wrench at a distance of 0.40m from the fulcrum. How 0.40m from the fulcrum. How

much torque must Ned produce much torque must Ned produce to turn the bolt?to turn the bolt?

Mabel and Maude are seesawing Mabel and Maude are seesawing on the school playground and on the school playground and

decide to see if they can move to decide to see if they can move to the correct location to make the the correct location to make the seesaw balance. Mabel weighs seesaw balance. Mabel weighs 400N and she sits 2m from the 400N and she sits 2m from the fulcrum of the seesaw. Where fulcrum of the seesaw. Where

should 450N Maude sit to balance should 450N Maude sit to balance the seesaw?the seesaw?

Moment of inertia:Moment of inertia:the resistance of an object to changes in its the resistance of an object to changes in its

rotational motionrotational motionHoop rotating about its center: I= mrHoop rotating about its center: I= mr22

Hoop rotating about its diameter: I= (1/2)mrHoop rotating about its diameter: I= (1/2)mr22

Solid cylinder: I= (1/2)mrSolid cylinder: I= (1/2)mr22

Stick rotating about its center of gravity: I= (1/12)mlStick rotating about its center of gravity: I= (1/12)ml22

Stick rotating about its end: I= (1/3)mlStick rotating about its end: I= (1/3)ml22

Solid sphere rotating about its center of gravity: I= (2/5)mrSolid sphere rotating about its center of gravity: I= (2/5)mr22

FormulaFormula

Angular momentum: the measure of how Angular momentum: the measure of how difficult it is to stop a rotating objectdifficult it is to stop a rotating object

Angular momentum (L)= (mass)(velocity)(radius)Angular momentum (L)= (mass)(velocity)(radius)

On the Wheel of Fortune game On the Wheel of Fortune game show, a contestant spins that show, a contestant spins that

15kg wheel that has a radius of 15kg wheel that has a radius of 1.40m. What is the moment of 1.40m. What is the moment of

inertia of this disk-shaped inertia of this disk-shaped wheel?wheel?

Trish is twirling her 0.60m Trish is twirling her 0.60m majorette’s baton that has a majorette’s baton that has a mass of 0.40kg. What is the mass of 0.40kg. What is the

moment of inertia of the baton moment of inertia of the baton as it spins about its center of as it spins about its center of

gravity?gravity?

Jupiter orbits the sun with a Jupiter orbits the sun with a speed of 2079m/s at an average speed of 2079m/s at an average

distance of 71,398,000m. If distance of 71,398,000m. If Jupiter has a mass of Jupiter has a mass of

1.90x101.90x102727kg, what is its angular kg, what is its angular momentum as it orbits?momentum as it orbits?