Circular MotionRotating Objects & Centripetal Acceleration

• Uniform circular motion causes an object to revolve around a single axis at a constant speed.

• Although the speed remains constant, the velocity does not.

ac

vt

• Note that the object shown in the example is always turning toward the left.This indicates that a force is pulling it toward the center of the rotation.

• Centripetal acceleration (ac) is experienced by objects in circular

motion, and is always directed toward the center of rotation.

ac

vt

• Where “r” is the radius of the circular path.• Centripetal force acts on an object in a circular path,

and is directed toward the center of rotation.

• ‘Centrifugal Force’ is an ‘apparent force’. It doesn’t actually exist.It is due to inertial effects.

Centripetal Acceleration:

• Centripetal force is not a natural force. It is simply the name given to a force directed toward the center of rotation.

Consider the two examples shown. What is the actual force behind the centripetal force

in each?

• Newton's 2nd Law can be applied to the concept of circular motion and centripetal force.

• F = ma Fc = mac

Centripetal Force:

• Arc length is represented by S.

• Radius is represented by r.• Angular displacement is

represented by .

• = S/r

Angular quantities of rotational motion:

Angular quantities of rotational motion:

• Angular speed ( ) is the angular displacement over time.

• The units are radians/second, or rad/s.• One revolution = 2π radians.

Angular quantities of rotational motion:• Angular acceleration is the change of angular

speed over time:• =Δω/Δt• The units are rad/s2

• Kinematics can be applied to rotation motion. • Simply replace the variables for linear velocity

and acceleration with those for angular speed and angular acceleration.

Relationships between angular & linear• Over small time intervals, the angular speed is nearly

identical to the tangential speed divided by the radius.• ω = v/r• Thus v = ωr.

The units reduce to m/s. Over a very small distance, there is essentially no change in

angle.

• Linear Kinematics can be rewritten to represent rotational motion.

• v ω• a • x

Chapter 7: Circular MotionNewtonian Gravitation

• Note that FG can be written as “mg”, where “m” represents one of the masses. Usually, m represents an object on a planet’s surface.

• Note that gravity can vary with altitude (r).

Law of Universal Gravitation

• The magnitude of the force of gravity can be calculated using Newton’s Law of Universal gravitation:

• Where G is the constant of universal gravitation.• G = 6.673 x 10-11 • The unit of G is

Law of Universal Gravitation

Chapter 7: Circular MotionTorque

• Consider object that rotates on a rigid axis…like a lever or a door.

• In order to move such an object, a force must be applied. The object will rotate around its axis in response to an applied force.

• Torque is a measure of the ability of a force to rotate an object.The farther a force is from the axis of rotation,

the easier it is to rotate the object.• Suppose you are pushing open a door.

Do you push/pull from the outer edge, or near the hinges?

Near the edge, obviously.It’s easier because it is farther from the hinges – the center of rotation.

• As the distance from the center of rotation increases, an applied force produces more torque.

• Where r is the “position vector” – the distance from the applied force (F) to the center of rotation.

• Torque has units of N*m (Newton-meters).

• Think of torque as “rotational work”.• In order to change an object’s rotational motion,

you must apply a torque.

Torque:τ = rF

• If multiple torques act on an object, the torques are added.If the net torque is not zero, then the object will rotate.

• The rate of rotation of an object does not change unless that object is acted on by a torque.

• In general, clockwise is negative. Counterclockwise is positive..

-+

EXAMPLE• Two psychology students are trying to use the same revolving door. They both push forward,

but cannot seem to figure out why the door isn’t working.• Student A pushes

forward 1.2 m from the center with a 625 N force.

• Student B pushesforward .80 m from the center with a force of 850 N.

• What is the net torque on the door? Will the psychology students be stuck in there forever? If not, which way will it rotate?

[-750 Nm + 680 Nm = -70 Nm / Clockwise]

BA

• What if the applied force is not perpendicular to the object?

• In such a case, the equation for torque becomes:

Where theta is the angle between the position vector (r) and the force (F).

Probably a good ideato copythis

Torque:τ = rFsinθ

• The relationship between torque and angular acceleration is expressed as:

• τ = mr2α• The value mr2 is known as the moment of

inertia, I.• Definition: moment of inertia - a body's

tendency to resist angular acceleration.

The larger the moment of inertia, the more torque required to accelerate it.Torque:

∑τ = Iα

Moment of inertia (point mass)I= mr2

• Why the ∑τ? Consider a rigid disk rotating about some

axis. The disk consists of many particles atdifferent distances from the axis.

The net torque is the sum of the torques provided by the individual particles. Because the disk is rigid, all particles have the same angular acceleration.

• An object’s moment of inertia depends on both its mass and the distribution of its particles.

• Conditions for Equilibrium:• The object must have a next external force equal to

zero:

• The next external torque must equal zero:

• Both conditions must be satisfied for equilibrium to be achieved.

• Keep in mind that the Forces and Torques may need to be evaluated in terms of X and Y.

∑F = 0

∑τ = 0