ROTATIONAL MOTION AND EQUILIBRIUM Angular Quantities of Rotational Motion Rotational Kinematics Torque Center of Gravity Moment of Inertia Rotational Kinetic Energy Angular Momentum Rotational Equilibrium

t q D = ANGULAR SPEED AND ANGULAR ACCELERATION: w = angular speed rad/s deg/s rev/s a = angular acceleration 2 rotating drum axis t q D =

ANGULAR CONVERSIONS: 1 rev = 2 p rad = 360 deg p rad = 180 deg Convert 246 o to radians. Convert 16.4 rev to degrees.

Convert 246 o to radians. Convert 16.4 rev to degrees.

ROTATIONAL KINEMATICS: Uniform angular acceleration: q = instantaneous angular position w = instantaneous angular speed

Through what angle does a wheel rotate if it begins from rest EXAMPLE: Through what angle does a wheel rotate if it begins from rest and accelerates at 3 o /s 2 for 15 seconds? Also calculate its final angular speed.

Through what angle does a wheel rotate if it begins from rest ANSWER: Through what angle does a wheel rotate if it begins from rest and accelerates at 3 o /s 2 for 15 seconds? Also calculate its final angular speed.

Calculate the torque produced by the force acting on the rod as shown. EXAMPLE: Calculate the torque produced by the force acting on the rod as shown. pivot F = 16N 57 o 1.25m

Calculate the torque produced by the force acting on the rod as shown. EXAMPLE: line of action Calculate the torque produced by the force acting on the rod as shown. pivot F = 16N 57 o 1.25m 1 - Draw Line of Action

Calculate the torque produced by d EXAMPLE: line of action Calculate the torque produced by d the force acting on the rod as shown. pivot F = 16N 57 o 1.25m 1 - Draw Line of Action 2 - Draw Moment Arm

Calculate the torque produced by d EXAMPLE: line of action Calculate the torque produced by d the force acting on the rod as shown. pivot F = 16N 57 o 1.25m 1 - Draw Line of Action 2 - Draw Moment Arm 3 - Calculate Moment Arm

t = F·d = 16N·1.05m = 16.8Nm EXAMPLE: Calculate the torque produced by line of action Calculate the torque produced by d the force acting on the rod as shown. pivot F = 16N 57 o 1.25m 1 - Draw Line of Action 2 - Draw Moment Arm 3 - Calculate Moment Arm 4 - Calculate Torque t = F·d = 16N·1.05m = 16.8Nm

t = F·d = 16N·1.05m = 16.8Nm EXAMPLE: Calculate the torque produced by line of action Calculate the torque produced by d the force acting on the rod as shown. Rotation will be counter-clockwise pivot F = 16N 57 o 1.25m 1 - Draw Line of Action 2 - Draw Moment Arm 3 - Calculate Moment Arm 4 - Calculate Torque 5 - See Rotation t = F·d = 16N·1.05m = 16.8Nm

When g is considered constant cg is usually referred to as center of mass.

EXAMPLE: Suppose the three masses in the diagram y Suppose the three masses in the diagram m = are fixed together by massless rods and are 4 kg at (0, 3m) 2 free to rotate in the xy plane about their r center of gravity. The coordinates of their 2 center of gravity are (1m, 1m) as calculated r r m = 3 kg at (4m, 0) in the previous example. 1 3 3 x Calculate the moment of inertia of these m = 5 kg at (0, 0) masses about this axis. 1

ANSWER: Suppose the three masses in the diagram y Suppose the three masses in the diagram m = are fixed together by massless rods and are 4 kg at (0, 3m) 2 free to rotate in the xy plane about their r center of gravity. The coordinates of their 2 center of gravity are (1m, 1m) as calculated r r m = 3 kg at (4m, 0) in the previous example. 1 3 3 x Calculate the moment of inertia of these m = 5 kg at (0, 0) masses about this axis. 1

EXAMPLE: A solid ball rolls down a 40o incline from a height of 4m without slipping. The radius of the ball is 20cm and it begins from rest. What is the linear speed of the ball at the bottom of the incline?

Continued on next slide ANSWER: A solid ball rolls down a 40o incline from a height of 4m without slipping. The radius of the ball is 20cm and it begins from rest. What is the linear speed of the ball at the bottom of the incline? Basic Equation Values and expressions for initial and final quantities Continued on next slide

Continues from previous slide Substitutions made to produce working equation Simplify and solve for v The speed of the ball does not depend on its mass nor on its radius. It even does not depend on the angle of the incline, just the height from which it starts. Do the same calculation with a cylinder and determine which will reach the bottom of the incline first, the sphere or the cylinder if released together.

EXAMPLE:. A merry-go-round on a playground has a radius or 1 EXAMPLE: A merry-go-round on a playground has a radius or 1.5 m and a mass of 225 kg. One child is sitting on its outer edge as it rotates 1 rev/s. If the mass of the child is 50 kg, what will the new rotation rate be when the child crawls half way to the center of the merry-go-round?

ANSWER:. A merry-go-round on a playground has a radius or 1 ANSWER: A merry-go-round on a playground has a radius or 1.5 m and a mass of 225 kg. One child is sitting on its outer edge as it rotates 1 rev/s. If the mass of the child is 50 kg, what will the new rotation rate be when the child crawls half way to the center of the merry-go-round?

An object in equilibrium has no linear and no angular accelerations An object in equilibrium has no linear and no angular accelerations. That does not mean it isn’t moving, just not accelerating. When an object is stationary it is said to be in static equilibrium. Many of the equilibrium problems here will be static equilibrium problems.

a wall at the left and supported at the right by a cable. The EXAMPLE: 30 o cable wall hinge 45 rod A uniform rod of length l = 4m and mass m = 75kg is hinged to a wall at the left and supported at the right by a cable. The cable is attached to the rod /4 from its right edge. The rod makes a 30 angle with the horizontal and the cable makes a 45 angle with the rod. Calculate the tension in the cable and the force exerted on the rod by the hinge. Solution on next several slides

STEP 1: cable wall T rod 45 30 hinge R W Add Force Vectors: T = Tension W = Weight R = Reaction

STEP 1: 165O cable wall 210O T rod 45 pivot 30 hinge R W Add Force Vectors: T = Tension W = Weight R = Reaction Add Needed Angles

STEP 1: 210O 165O T W R Add Force Vectors: T = Tension W = Weight R = Reaction Add Needed Angles Add Pivot and Moment Arms: dT = Moment Arm For Tension dW = Moment Arm For Weight

On the next slide these force equations will be used to solve for Rx and Ry.