Chapter 8Chapter 8Rotational Dynamics Rotational Dynamics

and Static Equilibriumand Static Equilibrium

8.1 Rotational motion8.1 Rotational motion

8.2 Torque8.2 Torque

8.3 Equilibrium and Center of 8.3 Equilibrium and Center of massmass

8.1 Rotational motion8.1 Rotational motion

8.2 Torque8.2 Torque

8.3 Equilibrium and Center of 8.3 Equilibrium and Center of massmass

Chapter 8.1 Rotational Chapter 8.1 Rotational MotionMotion

• The Physics of rotational motion is The Physics of rotational motion is analogous to the physics of linear analogous to the physics of linear motion. For example, the new motion. For example, the new concept of Torque and angular concept of Torque and angular acceleration are the rotational acceleration are the rotational analogs of force and acceleration analogs of force and acceleration (F=ma), Newton’s II law.(F=ma), Newton’s II law.

Chapter 8.1Chapter 8.1

• Rigid Body – an object whose size Rigid Body – an object whose size and shape do not change as it and shape do not change as it moves.moves.

Chapter 8.1Chapter 8.1

3 basic types of motions of a rigid body:3 basic types of motions of a rigid body:

•Translational motion – moving forwardTranslational motion – moving forward

•Rotational motion - rotateRotational motion - rotate

•Combination motion – rotate and move Combination motion – rotate and move forwardforward

Draw trajectory of 3 Draw trajectory of 3 motions.motions.

Chapter 8.1Chapter 8.1

Chapter 8.1Chapter 8.1• All points on a wheel rotate with the same All points on a wheel rotate with the same

angle angle θθ..

• Because of that, every point on rotating Because of that, every point on rotating rigid body has the same angular velocity rigid body has the same angular velocity 𝝎𝝎..

AB

arc or length

Practice problemPractice problem

• The disk in a computer disk drive The disk in a computer disk drive spin up to 5400 rpm in 2.00 s. What spin up to 5400 rpm in 2.00 s. What is the angular acceleration of the is the angular acceleration of the disk? At the end of 2.00 s, how disk? At the end of 2.00 s, how many revolutions has the disk many revolutions has the disk made?made?

11-9 The Vector Nature of Rotational Motion

The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives

the sign.

• Page 200, # 1-4Page 200, # 1-4

Definition of TorqueDefinition of Torque

Torque is defined as the tendency to produce a change in rotational motion.

Torque is defined as the tendency to produce a change in rotational motion.

Examples:Torque is a twist or turn that tends to produce rotation. Applications are Applications are found in many common found in many common tools around the home or tools around the home or industry where it is industry where it is necessary to turn, tighten or necessary to turn, tighten or loosen devices.loosen devices.

Torque is Determined by Three Torque is Determined by Three Factors:Factors:

• The The magnitudemagnitude of the applied force. of the applied force.

• The The directiondirection of the applied force. of the applied force.

• The The locationlocation of the applied force. of the applied force.

• The The magnitudemagnitude of the applied force. of the applied force.

• The The directiondirection of the applied force. of the applied force.

• The The locationlocation of the applied force. of the applied force.

20 N

Magnitude of force

40 N

The 40-N force produces twice the torque as does the

20-N force.

Each of the 20-N forces has a different

torque due to the direction of force. 20 N

Direction of Force

20 N

20 N20 N

Location of forceThe forces nearer the

end of the wrench have greater torques.

20 N20 N

Units for TorqueUnits for TorqueTorque is proportional to the magnitude of F and to the distance r from the axis.

Torque is proportional to the magnitude of F and to the distance r from the axis.

= Fr = Fr Units: Nm or lbft

6 cm

40 N

= (40 N)(0.06 m)

= 2.40 Nm, cw

= 24.0 Nm, cw = 24.0 Nm, cw

Direction of TorqueDirection of Torque

Torque is a vector quantity that has direction as well as magnitude.

Torque is a vector quantity that has direction as well as magnitude.

Turning the handle of a screwdriver clockwise

and then counterclockwise will

advance the screw first inward and then outward.

Direction of TorqueDirection of TorqueOnly 2 directions: counterclockwise

torques are positive and clockwise torques are negative.

Positive torque: Counter-

clockwise, out of page

cw

ccw

Negative torque: clockwise, into page

Line of Action of a ForceLine of Action of a Force

The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.

The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force.

F1

F2

F3Line of action

The Moment ArmThe Moment Arm

The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.

The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation.

F2

F1

F3

r

rr

Only the tangential component of force causes a torque:

This leads to a more general definition of torque:

Calculating TorqueCalculating Torque

• Read problem and draw a rough figure.Read problem and draw a rough figure.

• Extend line of action of the force.Extend line of action of the force.

• Draw and label moment arm.Draw and label moment arm.

• Calculate the moment arm if necessary.Calculate the moment arm if necessary.

• Apply definition of torque:Apply definition of torque:

• Read problem and draw a rough figure.Read problem and draw a rough figure.

• Extend line of action of the force.Extend line of action of the force.

• Draw and label moment arm.Draw and label moment arm.

• Calculate the moment arm if necessary.Calculate the moment arm if necessary.

• Apply definition of torque:Apply definition of torque:

Torque = force x moment arm

Example 1:Example 1: An An 80-N80-N force acts at the end of force acts at the end of a a 12-cm12-cm wrench as shown. Find the torque. wrench as shown. Find the torque.

• Extend line of action, draw, calculate r.

= (80 N)(0.104 m) = 8.31 N m

= (80 N)(0.104 m) = 8.31 N m

r = 12 cm sin 600 = 10.4 cm

r = 12 cm sin 600 = 10.4 cm

Alternate:Alternate: An An 80-N80-N force acts at the end of force acts at the end of a a 12-cm12-cm wrench as shown. Find the torque. wrench as shown. Find the torque.

Resolve 80-N force into components as shown.

Note from figure: rx = 0 and ry = 12 cm

= (69.3 N)(0.12 m) = 8.31 N m as before = 8.31 N m as before

positive

12 cm

11-9 The Vector Nature of Rotational Motion

A similar right-hand rule gives the direction of the torque.

Conservation of angular momentum means that the total angular momentum around any axis must be constant. This is why gyroscopes are so stable.

Page 203, #11-15Page 203, #11-15

Calculating Resultant Calculating Resultant TorqueTorque• Read, draw, and label a rough figure.Read, draw, and label a rough figure.

• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.

• Extend lines of action for each force.Extend lines of action for each force.

• Calculate moment arms if necessary.Calculate moment arms if necessary.

• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).

• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.

• Read, draw, and label a rough figure.Read, draw, and label a rough figure.

• Draw free-body diagram showing all Draw free-body diagram showing all forces, distances, and axis of rotation.forces, distances, and axis of rotation.

• Extend lines of action for each force.Extend lines of action for each force.

• Calculate moment arms if necessary.Calculate moment arms if necessary.

• Calculate torques due to EACH individual Calculate torques due to EACH individual force affixing proper sign. CCW (+) and force affixing proper sign. CCW (+) and CW (-).CW (-).

• Resultant torque is sum of individual Resultant torque is sum of individual torques.torques.

Page 205, #16-20Page 205, #16-20

Chapter 8.3Chapter 8.3Zero Torque and Static Zero Torque and Static

EquilibriumEquilibrium

Chapter 8.3Chapter 8.3• Static equilibrium occurs when an

object is at rest – neither rotating nor translating.

Center of MassCenter of Mass

• The center of mass, also called the The center of mass, also called the centroid or center of gravity, is the centroid or center of gravity, is the point of a body at which the force of point of a body at which the force of gravity can be considered to act gravity can be considered to act

The center of mass (CM) The center of mass (CM) is the point where all of is the point where all of the mass of the object is the mass of the object is concentrated.concentrated.

• When an object is supported at its When an object is supported at its center of mass there is no net center of mass there is no net torque acting on the body and it will torque acting on the body and it will remain in static equilibrium remain in static equilibrium

How to find the center of How to find the center of massmass

• If the object is uniform, for example If the object is uniform, for example a meter stick, the center of mass a meter stick, the center of mass will be at the exact geometric will be at the exact geometric center.center.

• If the object is If the object is irregularirregular in shape, in shape, the center of mass is always located the center of mass is always located closer to the more massive endcloser to the more massive end

This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.

• The degree of stability in an object's The degree of stability in an object's position depends on how must its position depends on how must its center of gravity will be changed if it center of gravity will be changed if it is moved. is moved.

• StableStable (It will be stable if the center (It will be stable if the center of gravity lies below the pivot of gravity lies below the pivot point. )point. )

• UnstableUnstable

• NeutralNeutral

See-SawsSee-Saws

We all

remember the fun see-saw of

our youth.

But what

happens if . . .

Balancing Unequal MassesBalancing Unequal Masses

Need:

M1 d1 = M2 d2

M1M2

d1 d2

If an extended object is to be balanced, it must be supported through its center of mass.

Center of Mass

For two objects:

The center of mass is closer to the more massive object.

Chapter 8.3: Equilibrium

Translational EquilibriumTranslational Equilibrium• An object is said to be in An object is said to be in

Translational Equilibrium Translational Equilibrium if and if and only if there is no resultant only if there is no resultant force. force.

• This means that the sum of all This means that the sum of all acting forces is zero.acting forces is zero.

In the example, the resultant of the three forces A, B, and C acting on the ring must

be zero.

In the example, the resultant of the three forces A, B, and C acting on the ring must

be zero.

A

C

B

Conditions for Static Conditions for Static EquilibriumEquilibrium

If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; we are free to choose the one that makes our calculations easiest.

Solving static's problemsSolving static's problems

• Choose one object at a time and draw Choose one object at a time and draw free-body diagramfree-body diagram

• Choose a convenient coordinate Choose a convenient coordinate systemsystem

• Write the equilibrium force equations Write the equilibrium force equations and torque equation. Pay attention to and torque equation. Pay attention to determine the lever arm for each force determine the lever arm for each force correctly and a sign of torque. correctly and a sign of torque.

• Solve these equations for unknown. Solve these equations for unknown. Three equations are maximum of 3 Three equations are maximum of 3 unknowns to be solved for.unknowns to be solved for.

Practice problemPractice problem

Determine the mass of the monkey. In the process, Determine the mass of the monkey. In the process, also calculate also calculate T1T1, , T2T2, , T3T3, , T4T4, , T5T5, , T6T6, and , and W2W2 if if W1W1 equals 350 N. equals 350 N.

CONCLUSION: Chapter 5ACONCLUSION: Chapter 5ATorque Torque