Circular Motion

Centripetal Force Key Question: Why does a roller coaster stay on a track upside down on a loop?

Centripetal Force We usually think of acceleration as a change in speed. Because velocity includes both speed and direction, acceleration can also be a change in the direction of motion.

Centripetal Force Any force that causes an object to move in a circle is called a centripetal force. A centripetal force is always perpendicular to an object’s motion, toward the center of the circle.

Centripetal Force Fc = mv2 r Mass (kg) Linear speed (m/sec) force (N) Fc = mv2 r Radius of path (m)

Calculate centripetal force A 50-kilogram passenger on an amusement park ride stands with his back against the wall of a cylindrical room with radius of 3 m. What is the centripetal force of the wall pressing into his back when the room spins and he is moving at 6 m/sec? 1) You are asked to find the centripetal force. 2) You are given the radius, mass, and linear speed. 3) The formula that applies is Fc = mv2 ÷ r. 4) Solve: Fc = (50 kg)(6 m/sec)2 ÷ (3 m) = 600 N

Centripetal Acceleration Acceleration is the rate at which an object’s velocity changes as the result of a force. Centripetal acceleration is the acceleration of an object moving in a circle due to the centripetal force.

Centripetal Acceleration Speed (m/sec) Centripetal acceleration (m/sec2) ac = v2 r Radius of path (m)

Calculate centripetal acceleration 1) You are asked for centripetal acceleration and a comparison with g (9.8 m/sec2). 2) You are given the linear speed and radius of the motion. 3) ac = v2 ÷ r 4) Solve: ac = (10 m/sec)2 ÷ (50 m) = 2 m/sec2 The centripetal acceleration is about 20% or 1/5 that of gravity. A motorcycle drives around a bend with a 50-meter radius at 10 m/sec. Find the motor cycle’s centripetal acceleration and compare it with g, the acceleration of gravity.

Centrifugal Force We call an object’s tendency to resist a change in its motion its inertia. An object moving in a circle is constantly changing its direction of motion. Although the centripetal force pushes you toward the center of the circular path... ...it seems as if there also is a force pushing you to the outside. This apparent outward force is called centrifugal force.

Centrifugal Force Centrifugal force is not a true force exerted on your body. It is simply your tendency to move in a straight line due to inertia. This is easy to observe by twirling a small object at the end of a string. When the string is released, the object flies off in a straight line tangent to the circle.

Torque The door is free to rotate about an axis through O There are three factors that determine the effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied

Torque, cont Torque, t, is the tendency of a force to rotate an object about some axis t = r F t is the torque F is the force symbol is the Greek tau r is the length of the position vector SI unit is N.m

Direction of Torque Torque is a vector quantity The direction is perpendicular to the plane determined by the position vector and the force If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

Multiple Torques When two or more torques are acting on an object, the torques are added As vectors If the net torque is zero, the object’s rate of rotation doesn’t change

General Definition of Torque The applied force is not always perpendicular to the position vector The component of the force perpendicular to the object will cause it to rotate

General Definition of Torque, cont When the force is parallel to the position vector, no rotation occurs When the force is at some angle, the perpendicular component causes the rotation

General Definition of Torque, final Taking the angle into account leads to a more general definition of torque: t = r F sin q F is the force r is the position vector q is the angle between the force and the position vector

Lever Arm The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin q

Right Hand Rule Point the fingers in the direction of the position vector Curl the fingers toward the force vector The thumb points in the direction of the torque

Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related