Circular Motion

Rotational Quantities A O r  dAdA A point on an object, located a distance r from a fixed axis of rotation, rotates in such a way that it travels a distance d along the circumference of a circle. The ratio of d to r is defined to be the angle  measured in radians. θ = d / r and will be referred to as the angular displacement

Rotational Quantities A O B dAdA  dBdB A point on an object, located a distance r from a fixed axis of rotation, rotates in such a way that it travels a distance d along the circumference of a circle. The ratio of d to r is defined to be the angle  measured in radians. θ = d / r and will be referred to as the angular displacement In a similar fashion to linear motion: is the angular velocity (  = v/r) is the angular acceleration (  = a/r)

Chart of the Quantities A simple set of relationships exists between the linear and angular quantities. Linear Quantities: Linear Kinematic Expressions Angular Quantities:Conversions: Angular Kinematic Expressions

Monster Trucks A car’s speedometer is set for the tire that is designed for use with that particular car. Suppose that you put tires on that are twice the radius of the normal tires. Does your speedometer: 1.Read a slower speed than is actually the case? 2.Read a higher speed than is actually the case? 3.Read the correct speed? 4.Doesn’t matter because you can outrun anything else on the road. The speedometer reads ω and converts to a speed using v = ωr. If, in reality, r is twice as big as expected, your speed (v) will be twice as big as the speedometer calculates.

Bicycle Wheel A bicycle tire, initially rotating at 12 rad/s, is brought to rest by applying a brake. The tire takes 4 seconds to come to a halt. What was the acceleration of the wheel and how many rotations did it make before stopping? Assume the acceleration is constant over the 4 seconds.

Uniform Circular Motion Types of motion we’ve discussed: Uniform Motion (constant speed) Uniform Acceleration (constant acceleration) Projectile Motion (2D combination of first two types) New type of motion: Uniform Circular Motion Object moves in circular path at a constant speed Object of mass m swings in circle of radius R and completes one revolution in time T. What is its velocity? Magnitude of velocity = speed = constant T is the period (time it takes for a complete revolution) Direction = ?

Direction of the Velocity Direction of is same as direction of. The mass moves from point A to point B with the change in displacement given by the red vector. This change in displacement divided by the time it takes to travel between points A and B is the average velocity of the mass and is equal to the velocity at point P, the point midway between A and B (blue vector).

Velocity Changes Directions As the mass moves around the circle the magnitude of the velocity (it's speed) remains constant while the direction of the velocity constantly changes, as shown in the diagram below. One more term to discuss: frequency. Number of revolutions per unit time Our equation for the speed of an object can be extended to include frequency: Angular velocity is also sometimes called angular frequency:

Centripetal Acceleration From Newton’s Principia An object shot at a low horizontal speed will land very close to the bottom of the mountain top from which it was shot. If the object is shot at an increasingly higher horizontal speed it will land farther and farther from the base of the mountain.

Newton envisioned a horizontal velocity at which the curvature of the projectile's trajectory matched the curvature of the earth. Centripetal Acceleration From Newton’s Principia An object shot at a low horizontal speed will land very close to the bottom of the mountain top from which it was shot. If the object is shot at an increasingly higher horizontal speed it will land farther and farther from the base of the mountain.

Centripetal Acceleration If it were not for gravity, the object would travel in a straight line (at constant speed) and reach point X in time t. The distance that it would travel would be given by the horizontal motion equation: Due to gravity, the object arrives at point A having fallen a distance y given by the vertical equation: From the Pythagorean Theorem 2 R R y = ½ at x x = v t x A If t is allowed to go to zero (instantaneous acceleration), then we have What is direction?

Centrifuge The centrifuge is spinning so that the bottom of the test tube has a tangential speed of 90 m/s. If the test tube is 8.5cm long, what acceleration is experienced at the bottom of the tube? How many rotations does it make per minute?

Dynamics of Circular Motion Newton's 2 nd Law provides us the insight we need to explain why circular motion occurs. An object accelerates toward the center of a circle due to the action of a net force in that direction. This force is referred to as the Centripetal Force. THIS IS NOT A NEW FORCE! Can simultaneously have tangential forces.

Vertical Circles A cup supported by a platform and suspended by strings is being swung in a vertical circle. Take a look at the forces at each of the indicated positions. At the 3 and 9 o'clock position there are two forces acting on the cup. The weight is acting straight down and does not contribute to the circular motion of the cup. The normal force is the only force pointing toward the center of the circle and must be responsible for the centripetal acceleration. At the 6 and 12 o'clock position the weight and the normal force are in the same dimension and must together add up to be responsible for the centripetal acceleration.

Limiting Case Consider object at top. What is the normal force? What does it mean physically if F N = 0? Minimum speed required to not fall off and turn into a projectile! Independent of mass.