Unit 8

Center of Mass A point that represents the average location for the total mass of a system For symmetric objects, made from uniformly distributed material Center of mass = Geometric center

Center of Mass & Motion Center of mass follows a projectile path

Rotational Motion Focuses on pure rotational motion Motion that consists of rotation about a fixed axis Points on a rigid object move in circular paths around an axis of rotation Examples: Ferris Wheel; CD

Angular Position In the study of rotational motion, position is described using angles (θ) Angular Position = The amount of rotation from a reference point Counterclockwise rotation = positive angle Clockwise rotation = negative angle Units = radians

Angular Position For a full revolution:

Angular Displacement In the study of rotational motion, angular displacement is also described using angles (θ) Change in angular position Δθ = θ f − θ i Δθ = angular displacement θ f = final angular position θ i = initial angular position Units = radians (rad) Counterclockwise rotation = Positive Clockwise rotation = Negative

Angular Displacement The angle through which the object rotates is called the angular displacement.

PROBLEM: Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is 4.23×10 7 m. If the angular separation of the two satellites is 2 degrees, find the arc length that separates them.

ANSWER

How we describe the rate at which angular displacement is changing… Angular Velocity

SI Unit of Angular Velocity: radian per second (rad/s) Average Angular Velocity

Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time of 1.90 s. Find the average angular velocity of the gymnast. Problem

Answer

Problem It takes a motorcycle rider 2 seconds to make a counterclockwise lap around a track. What is his average angular velocity?

Changing angular velocity means that an angular acceleration is occurring. DEFINITION OF AVERAGE ANGULAR ACCELERATION SI Unit of Angular acceleration: radian per second squared (rad/s 2 ) Angular Acceleration

A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant. Problem

Answer

Problem Over the course of 1 hour, what is (a) The angular displacement (b) The angular velocity and (c) The angular acceleration of the minute hand?

Answer VARIABLES: Elapsed Time Δt = 1 h  3600 s Angular DisplacementΔθ Angular Velocityω Angular Accelerationα STEP-BY-STEP SOLUTION * Angular Displacement Δθ = -2πrad (Minute hand travels clockwise one revolution) Δt = 1 h  3600 s * Angular Velocity ω = Δθ/Δt = -2πrad/3600s = -((2)(3.14))/3600 = x rad/s * Since the angular velocity is constant, angular acceleration is zero

Problem At a particular instant, a potter's wheel rotates clockwise at 12.0 rad/s; 2.50 seconds later, it rotates at 8.50 rad/s clockwise. Find its average angular acceleration during the elapsed time.

Answer Average Angular Acceleration: α = Δω / Δt = (-8.5 rad/s) – (-12 rad/s) / 2.5 s = 1.4 rad/s 2

Did NOT do slides beyond this point

Angular Velocity Vector Angular Velocity Vector: * Parallel to the axis of rotation * Magnitude of the angular velocity vector is proportional to the angular speed - Faster the object rotates  longer the vector * Direction of the angular velocity vector determined by the right-hand rule

Right-Hand Rule Right-hand Rule: * Used to determine the direction of the angular velocity vector * Fingers of the right hand curl in the direction of rotation * Thumb then points in the direction of the angular velocity vector

Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity. Angular Velocity Vector

Angular Acceleration Vector Angular Acceleration Vector: * If the object is speeding up  its angular velocity vector is increasing in magnitude  Angular acceleration vector points in the same direction as the angular velocity vector * If the object is slowing down  its angular velocity vector is decreasing in magnitude  Angular acceleration vector points in the opposite direction as the angular velocity vector Preview Kinetic Books: 10.15

Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, v o 5. elapsed time, t Recall… The Kinematic Equations For Constant Acceleration

ANGULAR DISPLACEMENT ANGULAR VELOCITY ANGULAR ACCELERATION TIME Rotational Kinematic Equations: For Constant Angular Acceleration

Rotational Kinematic Equations

1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem. Reasoning Strategy

PROBLEM: Blending with a Blender The blades are whirling with an angular velocity of +375 rad/s when the “puree” button is pushed in. When the “blend” button is pushed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of rad. The angular acceleration has a constant value of rad/s 2. Find the final angular velocity of the blades.

θαωωoωo t rad+1740 rad/s 2 ?+375 rad/s

Tangential Velocity Tangential Velocity: * Linear velocity at an instant - Magnitude = magnitude of linear velocity - Direction: Tangent to circle * v T = rω v T = tangential speed r = distance to axis ω = angular velocity Direction: tangent to circle Preview Kinetic Books: 10.11

Tangential Velocity

Tangential Acceleration Tangential Acceleration: - Rate of change of tangential speed - Increases with distance from center - Direction of vector is tangent to circle - a T = rα a T = tangential acceleration r = distance to axis α = angular acceleration Preview Kinetic Books: 10.12

Tangential Acceleration

PROBLEM: A Helicopter Blade A helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s 2. For point 1 on the blade, find the magnitude of (a) The tangential speed (b) The tangential acceleration.

Centripetal Acceleration

Problem: A Discus Thrower Starting from rest, the thrower accelerates the discus to a final angular speed of rad/s in a time of s before releasing it. During the acceleration, the discus moves in a circular arc of radius m. Find the magnitude of the total acceleration.

Centripetal Acceleration & Tangential Acceleration

Rotational Motion Review UNIFORM CIRCULAR MOTION: Example- Spinning Ferris wheel or an orbiting satellite Object moves in a circular path and at a constant speed The object is accelerating, however, because the direction of the object’s velocity is constantly changing Centripetal acceleration  Directed toward the center of the circle Net force causing the acceleration is a centripetal force  Also directed toward the center of the circle In this section, we will examine a related type of motion  The motion of a ROTATING RIGID OBJECT

The Motion of a Rotating Rigid Object A football spins as it flies through the air If gravity is the only force acting on the football  Football spins around a point called its center of mass As the football moves through the air, its center of mass follows a parabolic path

Rotational Dynamics Study of Rotational Kinematics Focuses on analyzing the motion of a rotating object Determining such properties as its angular displacement, angular velocity or angular acceleration Study of Rotational Dynamics Explores the origins of rotational motion * Preview Kinetic Books 11.1

Torque Every time you open a door, turn of a water faucet, or tighten a nut with a wrench  you are exerting a TURNING FORCE Turning force produces a TORQUE Review… If you want to make an object move  Apply a force Forces make things accelerate If you want to make an object turn or rotate  Apply a torque Torques produce rotation

Torque A force that causes or opposes rotation The amount of torque depends on… The amount of force When more force is applied  there is more torque The distance from the axis of rotation to the point of force

The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

Torque For a force applied perpendicularly τ = magnitude of torque (Greek letter tau) r = distance from axis to force F = force Direction: Torque resulting from a force is + if the rotation is counterclockwise and – if the rotation is clockwise Counterclockwise +, clockwise − S.I. Units: newton-meters (N∙m)

Torque Imagine a dog-flap door Rotates on a hinge Allowing pets to enter and leave a house at will Axis of Rotation Force: * If the dog pushed on the door with the same force but at a point closer to the hinge, the door would be more difficult to rotate Lever Arm: Perpendicular distance from the axis of rotation to a line draw along the direction of the force Lever Arm

Torque, angle & lever arm * Preview Kinetic Books 11.2 Forces do not have to be perpendicular to an object to cause the object to rotate τ = r × F τ = rF sin θ τ = Torque (N x m)  Vector quantity r = Position vector (m) (AKA: Displacement) F = Force (N)  Vector quantity θ = angle between r and F

Calculating Torque using the Lever Arm Torque can also be calculated using the concept of a lever arm Lever arm is the perpendicular distance from the axis of rotation to the line containing the force vector “Line of action” τ = F (r sin θ) * r sin θ = lever arm Mathematically: τ = F (r sin θ) is the same as τ = rF sin θ

Torque

Problem A basketball is being pushed by two players during a tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11 N at a perpendicular distance of 7 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass.

Answer Given: F 1 = 15Nr 1 = 0.14m F 2 = 11Nr 2 = 0.07m τ net = ? τ = F 1 r 1 + F 2 r 2 (** The factor sin Θ is not included because each distance is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force)

Answer Continued τ = F 1 r 1 + F 2 r 2 * Use the standard convention of signs… τ = (15N)(-0.14m) + (-11N)(0.070m) τ = -2.9 Nxm Net torque is negative, so the ball rotates in a clockwise direction

The Achilles Tendon The tendon exerts a force of magnitude 790 N. Determine the torque (magnitude and direction) of this force about the ankle joint.

790 N Answer τ = rF sin θ