Circular Motion Uniform Circular Motion is rotational motion where the angular velocity of an object is constant. Because we are moving in circles and changing the direction of the velocity vector, there must be an acceleration present, even though our speed remains constant. We will discuss this further, later in this unit. The key to understanding all types of circular motion (uniform and accelerated) is to understand that all of the circular variables have the same relationships as their linear counterparts.

Understanding angles: Degrees and Radians Degrees and Radians are two types of angular measurement. 1 rad = approx. 57.3o 1o = approx. 1.74 x 10-2 rad To convert, set up a ratio where These are approximations because of the  factor in the conversion.

Angular displacement Roughly, displacement refers to how far something moves from its starting position. Since revolutions in a circle or on a wheel would continue to bring a reference point on the circle back to the same spot (x=0) angular displacement refers to the quantity of angles the reference point has swung through. The symbol theta () is used for angular displacement.

Sometimes this is also called “s,” the “arc length.”

Angular velocity: How fast something spins. Linear velocity is equal to change in displacement over change in time. Unfortunately, linear velocity, as it relates to displacement is not terribly meaningful when describing movement in circles. Why do you suppose this is?

Angular velocity describes how fast something spins and thus must be related to angular displacement. Angular velocity is equal to the net change in angular position divided by the time taken making that change.  = /t

Angular acceleration Angular acceleration, like linear acceleration, refers to how fast a velocity changes. Angular acceleration is equal to the change in angular velocity over time.  = /t

Linear and Angular Motion What’s up with this “t” going on here? This refers to the tangential component of the acceleration. That is to say the acceleration that acts around the circle.

Relationships:  = /t  = /t  = o + t  = o +t  = ot + ½t2 2 - o2 = 2 Look familiar?

Now it gets icky... The period of revolution is the amount of time it takes the rotating object to make one complete revolution. When you know the linear components. When you know the angular components.

Centripetal Acceleration A rotating body is in many respects like a projectile (a projectile with a high enough horizontal velocity is a satellite, an object that orbits, or rotates around the earth). Rotating bodies have a component of acceleration that constantly pulls the object along the radius towards the center of the circle or orbit. This is the centripetal acceleration. Centripetal acceleration, since it is perpendicular to the velocity vector at all times, does not change the magnitude of said vector, only the direction. The concept is similar to the change in direction that a projectile undergoes during flight.

The centripetal acceleration, ac, ensure that the object continues to “fall”towards the center of the circle as it rotates. Tangential Angular The tangential acceleration, at, and the tangential velocity and displacement relate respectively to the object’s acceleration, velocity and displacement around the circle.

Centripetal Force Centripetal force is simply to force associated with and causing the centripetal acceleration, ac. Any object traveling in a circle or a circular path experiences a centripetal force. The car problem: You are in the backseat of your parents sedan. When the car turns sharply to the right (as in making a 90o turn at a corner) your body is forced to the left. If a centripetal force is actually pulling you and the car radially towards the inside of the turn, why do you feel a force pushing you away from the center of rotation? (hint: think about Newton’s 1st Law!)

Centripetal Force: A NET Force The centripetal force is, in physics, what we call an imaginary force. This does NOT mean that the force is non-existent, but rather this means that the force we identify mathematically as “centripetal” is ALWAYS caused by some other force. E.g., a contact (normal) force, a friction or a tension. This means that mv2/r is always equal to something else that you must define based on the circumstances of the system you are studying.

Torque Torque is simply a force applied to an object at some radial distance from the center of rotation. If the torque is large enough to overcome the rotational inertia of the object it will cause the object to rotate. The “radial distance” is sometimes called the lever arm. Torque, like force, is governed by Newton’s Second Law:

Torques work in equilibrium just like forces, Radius Torque: Lower case “tau” Angular acceleration Moment of inertia: describes the arrangement of mass around a central point, often something you look up... Torques work in equilibrium just like forces, and all of the same rules apply. E.g., When the net torque on an object is zero it is either resting, or rotating with constant rotational velocity.

The torque used to create or sustain a rotation is equal to the applied force multiplied by the radius at which that force is applied if, and only if, the force is applied such that is is perpendicular to the radius. We will make this assumption in the course when solving problems unless otherwise stated.

Some problems for your mental stimulation pleasure... 1) A wheel rotates at a rate of 8 rad/s for 16 s. What is the angular displacement of the wheel? How many revolutions as it made? 2) If the wheel in problem 1 is actually a car tire with a diameter of 0.72 m (roughly 32 inches), how far has the car traveled? 3) a) Calculate the centripetal acceleration on a roller coaster car of mass m = 100 kg that goes through a circular loop of radius 25m at 55 m/s. b) what is the centripetal force on the car and occupants? c) what is the angular velocity of the roller coaster? 4. A 0.50 m crowbar is used to rip open a wooden crate. What force is necessary to apply to the crowbar if the torque necessary to strip boards off the crate is 47 Nm? 5) If the angular acceleration caused by the torque in problem 4 is 0.2 rad/s2, what is the moment of inertia for the crowbar?