Circular MotionCircular Motion

Physics

Mr. Padilla

Rotation and RevolutionRotation and Revolution

Both rotation and revolution occur by an object turning about an axis.

Rotation - The axis is located within the rotating body.– Ex: Merry-go-round

Revolution - Object turns around an external axis. – Ex: Earth around the sun

Angles can be measured in radians

In science, angles are often measured in radians (rad) rather than degrees.

θ = s

rs = arc lentghr = length of radius

Converting from Degrees to Radians

θ (rad) = __ π__ θ (deg)

180o

SpeedSpeedSpeed of a rotating object can be measured

in a few different ways.Linear speed - distance per unit time

– A.K.A. - Tangential Speed (m/s)

Rotational speed - (angular speed) rate at which an object rotates– Rotations per minute (RPM)– Radians/second

More SpeedMore Speed

Two people sit on a merry-go-round, one on the outside, one on the inside.

Which one has a greater tangential speed?– The one on the outside

Which has a greater rotational speed?– They both have the same angular speed

θ is analogous to x

Angular DisplacementThe angle through which a point line, or

body is rotated in a specified direction and about a specified axis.

Can be found by the change in arc length divided by the distance from the axis of rotation.

∆Ө = ∆s

r

Angular Displacement

∆Ө is positive when the object rotates counterclockwise

∆Ө is negative when the object rotates clockwise.

Sample Problem 7A

While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is 165o, what is the radius of the carousel?

Angular SpeedAngular SpeedAngular speed is a measure of how fast an

object moves in a circleAverage angular speed = angular

displacement / timeӨ is angular displacementω = ∆Ө/∆t

– Ө is measured in radians– Number of rotations * 2π = Ө (in radians)– ω is analogous to v

Sample Problem 7B

A child at an ice cream parlor spins on a stool. The child turns counterclockwise with an average angular speed of 4.0 rad/s. In what time interval will the child’s feet have an angular displacement of 8.0π rad?

Angular Acceleration

The time rate of change of angular speed, expressed in radians per second squared.

αavg = ∆ = 2 - 1

∆t t2 – t1

Angular Acceleration ExampleAngular Acceleration Example

What α is needed to increase the speed of a fan blade from 8.5rad/s to 15.4rad/s in 5.2s?

αavg = Δω/t

αavg = (15.4rad/s-8.5rad/s)/5.2s

αavg = 1.3 rad/s2

Sample Problem 7C

A car’s tire rotates at an initial angular speed of 21.5 rad/s. The driver accelerates, and after 3.5 s the tire’s angular speed is 28.0 rad/s. What is the tire’s average angular acceleration during the 3.5 s time interval?

Practice AssignmentPractice Assignment

Practice Q’s:

7A-C odd

Rotational KinematicsRotational Kinematics

Linear

vf = vi + at

Δx = ½ at2 + vit

vf2 = vi

2 + 2aΔx

Δx = ½ (vi + vf)t

Rotational

ωf = ωi + αt

Δθ = ½ αt2 + ωit

ωf2 = ωi

2 + 2αΔθ

Δθ = ½ (ωi + ωf)t

Sample Problem 7D

The wheel on an upside-down bicycle moves through 11.0 rad in 2.0 s. What is the wheel’s angular acceleration if its initial speed is 2.0 rad/s?

Tangential SpeedTangential Speed

Tangential speed ~ radial distance x rotational speed.

The farther out an object is, the faster its tangential speed.

vt = r is in rad/s rot/s * 2π = rad/s

If object A is twice as far from the axis of rotation as object B, how much faster is object A going?

Object A will be going twice as fast.

ProblemsProblems A wheel rotates at 90

rpm. What is the tangential speed of an object .3m from the center?– First: 90 rpm = 1.5rot/s– 1.5rot/s * 2π =

9.42 rad/s vt = r vt = (.3m)(9.42 rad/s) vt = 2.83m/s

A stone in a sling moves with a linear velocity of 6m/s. If the sling is .5m metes in length, how fast is it being twirled?

v = r 6 m/s = (.5m) = 12 rad/s

Sample Problem 7E

The radius of a CD in a computer is 0.0600 m. If a microbe riding on the disc’s rim has a tangential speed of 1.88 m/s, what is the disc’s angular speed?

Tangential Acceleration

The instantaneous linear acceleration of an object directed along the tangent to the object’s circular path.

α must be in rad/s2

at = rα

Sample Problem 7F

A spinning ride at a carnival has an angular acceleration of 0.50 rad/s2. How far from the center is a rider who has a tangential acceleration of 3.3 m/s2?

Centripetal ForceCentripetal Force

Def: Any force that causes an object to follow a circular path.– When you swing a ball around on a string, the

string provides the centripetal force.– When the Earth revolves around the sun,

gravity provides the centripetal force.

Centripetal force is always pulling towards the center of the circle

Centripetal AccelerationCentripetal Acceleration

Acceleration directed toward the center of a circular path

Centripetal Acceleration ac = vt2

ror

ac = rω2

Sample Problem 7G

A test car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s2, what is its tangential speed?

Finding Total Accelerationat

atotalac

atotal2 = at

2 + ac2

Causes of Circular Motion

A ball of mass m is tied to a string of length r that is whirled in a horizontal circular path. Assume the ball moves with a constant speed.

Does the ball experience centripetal acceleration?

The ball does experience centripetal acceleration because the velocity vector changes direction continuously during the motion.

The ball also experience a change in force.

Centripetal Force

Newton’s 2nd Law can also be applied along the radial direction

Fc = mac or Fc = mvt2 or Fc = mr2

r

ExampleExample

Calculate the force needed to keep a 1000kg car going in a circle with a radius of 10m, if the car’s velocity is 40m/s

What is providing the force?

ac = v2/r

– ac = (40m/s)2/10m

– ac =160m/s2

Fc = mac

– F = (1000kg)(160m/s2)– F = 160,000 N

Friction

Sample Problem 7H

A pilot is flying a small plane at 30.0 m/s in a circular path with a radius of 100.0 m. If a force of 635 N is needed to maintain the pilot’s circular motion, what is the pilot’s mass?

CentrifugalCentrifugal

Means “center fleeing”A common misconception is that when you

swing the ball around on the string, the centrifugal force will cause it to fly out from its path.– What really causes the ball to go outward?

Centrifugal force is considered to be a fictitious force.

More CentrifugalMore Centrifugal

What is really causing an object to be “pulled” outward? …its inertia

From an outside perspective, this is not a real force, but from a rotating frame of reference, this force appears real.

This appearance can be felt as simulated gravity.

Simulated GravitySimulated Gravity

The amount of simulated gravity depends on how fast things rotate and the radius of rotation.

If the object spinning is to small, it would have to rotate very quickly to get an a = g

Too fast of an acceleration could make a person dizzy or sick

About 1 rpm is acceptable for most people.

For a space station to experience simulated gravity at 1 rpm, it would have to be very large.

People at the outer edge would experience a = g, those closer would have a less than g.– Half way = .5g– Center = 0g