Chapter 8Chapter 8

Rotational Motion and Rotational Motion and EquilibriumEquilibrium

Rigid BodyRigid Body

An object or system of particles in An object or system of particles in which the distances between which the distances between particles are fixed.particles are fixed.

Translational (sliding) and Translational (sliding) and rotational (spinning) motion rotational (spinning) motion becomes relevant when we becomes relevant when we consider rigid bodies. consider rigid bodies.

Translational vs Translational vs Rotational MotionRotational Motion

Translational Motion: Every Translational Motion: Every particle in the rigid body has the particle in the rigid body has the same instantaneous velocity (no same instantaneous velocity (no rotation).rotation).

Rotational Motion: Every particle Rotational Motion: Every particle in the rigid body has the same in the rigid body has the same angular velocity and travels in angular velocity and travels in circles around a fixed axis.circles around a fixed axis.

Rotation and Rotation and TranslationTranslation

Rolling MotionRolling Motion

Rolling without slipping is a Rolling without slipping is a combination of rotation and combination of rotation and translation.translation.

TorqueTorque

A force is necessary to produce changes in A force is necessary to produce changes in rotational motion as well as translational. rotational motion as well as translational.

In pure translational motion, the translational In pure translational motion, the translational acceleration is proportional to the net force. Facceleration is proportional to the net force. Fnetnet = ma= ma

In pure rotational motion, the angular In pure rotational motion, the angular acceleration is related to the net force AND to acceleration is related to the net force AND to the the perpendicular distance between the axis of perpendicular distance between the axis of rotation and the line of actionrotation and the line of action. We call this . We call this distance the lever arm or rdistance the lever arm or r

Torque TerminologyTorque TerminologyLine of Action – an imaginary line Line of Action – an imaginary line

extending through the force extending through the force vectorvector

Lever Arm – r - the perpendicular Lever Arm – r - the perpendicular distance from the axis of rotation distance from the axis of rotation to the line of actionto the line of action

Line of Action and Lever Line of Action and Lever ArmArm

Line of Action and Lever Line of Action and Lever ArmArm

TorqueTorque

Torque = (lever arm Torque = (lever arm distance)distance)·(Force)·(Force)

Torque = r ·FTorque = r ·F

Torque = r F sinTorque = r F sinθθ

TT = = rr FF sin sin θθ

Torque DirectionTorque Direction

The right hand rule can be used to find The right hand rule can be used to find the direction of Torque as r X Fthe direction of Torque as r X F

TT = = rr x x FF

Alternatively, torques that cause Alternatively, torques that cause counterclockwise rotation are taken to counterclockwise rotation are taken to be positive and torques causing be positive and torques causing clockwise rotation are taken to be clockwise rotation are taken to be negative.negative.

TorqueTorque TT = = rr FF sin sin θθ

Torque is a vector, r is a vector Torque is a vector, r is a vector and F is a vector!and F is a vector!

Torque is measured in [m·N]Torque is measured in [m·N] The unit for Torque is NOT The unit for Torque is NOT

equivalent to a Joule [N·m]equivalent to a Joule [N·m] Right hand rule can be used to Right hand rule can be used to

find the direction of torque.find the direction of torque.

ExamplesExamples

When you lift up on something When you lift up on something with your forearm, torque is with your forearm, torque is applied on the lower arm by the applied on the lower arm by the biceps muscle. With the axis of biceps muscle. With the axis of rotation through the elbow joint rotation through the elbow joint and the muscle attached 4.0 cm and the muscle attached 4.0 cm from the joint, what is the torque from the joint, what is the torque if the muscle exerts 600N of if the muscle exerts 600N of force?force?

EquilibriumEquilibrium Balanced forces create translational Balanced forces create translational

equilibrium.equilibrium.

Balanced torques create rotational Balanced torques create rotational equilibrium.equilibrium.

Mechanical equilibrium occurs when Mechanical equilibrium occurs when forces and torques balance.forces and torques balance.

Static equilibrium occurs when a rigid Static equilibrium occurs when a rigid body is at rest and forces and torques body is at rest and forces and torques balance.balance.

ExampleExample A picture hangs motionless on a A picture hangs motionless on a

wall. The picture has a mass of wall. The picture has a mass of 3.0 kg. Find the magnitude of the 3.0 kg. Find the magnitude of the tension in the wires if one wire tension in the wires if one wire makes a 50makes a 50° angle and the other ° angle and the other wire makes a 45° angle.wire makes a 45° angle.

Example: Rotational Example: Rotational EquilibriumEquilibrium Three masses are suspended from Three masses are suspended from

a meter stick balanced at the a meter stick balanced at the halfway point . 25g hangs from 0 halfway point . 25g hangs from 0 cm and 75g hangs from 20cm. cm and 75g hangs from 20cm. Unknown mass hangs at 85cm. Unknown mass hangs at 85cm. Neglecting the mass of the meter Neglecting the mass of the meter stick, how much mass must be stick, how much mass must be suspended on the right side for the suspended on the right side for the system to be in static equilibrium?system to be in static equilibrium?

ExampleExample

A ladder with a mass of 15 kg A ladder with a mass of 15 kg rests against a smooth wall. A rests against a smooth wall. A painter who has mass 78 kg painter who has mass 78 kg stands on the ladder as shown. stands on the ladder as shown. What is the magnitude of the What is the magnitude of the frictional force that acts on the frictional force that acts on the bottom to keep the ladder from bottom to keep the ladder from sliding?sliding?