**Uniform Circular Motion

Introduction to Uniform Circular Motion

Uniform Circular Motion An object moves at uniform speed in a circle of constant radius. Is an object in uniform circular motion accelerating?

Acceleration in Uniform Circular Motion The velocity vector at any given point is subjected to an acceleration that turns it, but does not speed it up or slow it down. The acceleration vector is always at right angles to the velocity. The acceleration points toward the center of the circle.

r

v r velocity is always tangent to the circle

v r So according to Newton, an object will remain in motion unless acted on by a force. So where is the force?

v Fc r Centripetal Force always points towards the center of the circle.

v Fc r So according to Newton, unbalanced force creates an acceleration. Where is the acceleration?

v Fc ac r Centripetal Acceleration always points towards the center of the circle.

v Fc ac r An acceleration is a change in velocity per unit time. So if there is an acceleration then the velocity must change.

v r ∆v The change in velocity happens in the direction of the acceleration.

v r ∆v But the change in velocity happens over time. In some time in the future the object has moved forward.

vi ∆v vi r ∆v But the change in velocity happens over time. In some time in the future the object has moved forward.

∆v vi vi r ∆v An we can use what we know about vector addition to find the new velocity vector. Side the vectors, tip to tail and find the resultant.

∆v vf vi vi r ∆v An we can use what we know about vector addition to find the new velocity vector. Side the vectors, tip to tail and find the resultant.

vf vi r So we see that our initial velocity and final velocity have the same magnitude but a new direction.

Force in Uniform Circular Motion A force responsible for uniform circular motion is referred to as a centripetal force. Centripetal force is simply mass times centripetal acceleration. Fc = mac

Centripetal Force Fc = m ac v Fc: centripetal force in N v: tangential speed in m/s r: radius in meters Fc v Fc Fc v Always toward center of circle!

Acceleration in Uniform Circular Motion The acceleration responsible for uniform circular motion is referred to as centripetal acceleration.

Centripetal Acceleration v ac = v2/r ac: centripetal acceleration in m/s2 v: tangential speed in m/s r: radius in meters ac v ac ac v Centripetal acceleration always points toward center of circle!

Centripetal Force v Now that we know centripetal acceleration we can write centripetal force another way. Fc = m ac or Fc = m v2 / r Fc v Fc Fc v Always toward center of circle!

So what happens if the string breaks?

A 1x103 kilogram car travels at a constant speed of 20 A 1x103 kilogram car travels at a constant speed of 20. meters per second around a horizontal circular track. The diameter of the track is 1x102 meters. The magnitude of the car's centripetal acceleration is 8 m/s2

A 0.50-kilogram object moves in a horizontal circular path with a radius of 0.25 meter at a constant speed of 4.0 meters per second. What is the magnitude of the object’s acceleration? 64 m/s2

An unbalanced force of 40. newtons keeps a 5 An unbalanced force of 40. newtons keeps a 5.0-kilogram object traveling in a circle of radius 2.0 meters. What is the speed of the object? 4 m/s

A 1750-kilogram car travels at a constant speed of 15 A 1750-kilogram car travels at a constant speed of 15.0 meters per second around a horizontal, circular track with a radius of 45.0 meters. The magnitude of the centripetal force acting on the car is 8750 N

Universal Gravitation The gravitational force is a pulling force felt by all objects with mass on each object. The gravitational force depends on the mass of each object and the distance between them.