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Chapter 8

Rotational Equilibrium and

Rotational Dynamics

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Force vs. Torque

Forces cause accelerationsTorques cause angular accelerationsForce and torque are related

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Torque

The door is free to rotate about an axis through OThere are three factors that determine the effectiveness of the force in opening the door:

The magnitude of the forceThe position of the application of the forceThe angle at which the force is applied

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Torque, contTorque, τ, is the tendency of a force to rotate an object about some axis

τ = r F τ is the torqueF is the force

symbol is the Greek tau

r is the length of the position vector

SI unit is N.m

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QQ1Using a screwdriver, you try to remove a screw from a piece of furniture, but can’t get it to turn. To increase the chances of success, you should use a screwdriver that (a) is longer, (b) is shorter, (c) has a narrower handle (d) has a wider handle.

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Direction of TorqueTorque is a vector quantity

The direction is perpendicular to the plane determined by the position vector and the forceIf the turning tendency of the force is counterclockwise, the torque will be positiveIf the turning tendency is clockwise, the torque will be negative

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Right Hand Rule

Point the fingers in the direction of the position vector (beginning from the axis of rotation)Curl the fingers toward the force vectorThe thumb points in the direction of the torqueOut of the page for the figure

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Multiple Torques

When two or more torques are acting on an object, the torques are added

As vectors. Take sign into account

If the net torque is zero, the object’s rate of rotation doesn’t change

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General Definition of Torque

The applied force is not always perpendicular to the position vector

When the force is parallel to the position vector, no rotation occursWhen the force is at some angle, the perpendicular component causes the rotation

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General Definition of Torque, final

Taking the angle into account leads to a more

general definition of torque:

τ = r F sin θF is the forcer is the position vectorθ is the angle between the force and the position vector

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Lever Arm

The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the forced = r sin θFormula for torque may be rearranged to show that the torque is proportional to the force F times lever arm

τ = r F sin θ = F (r sin θ) = F d

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Net TorqueThe net torque is the sum of all the torques produced by all the forces

Remember to account for the direction of the tendency for rotationwhich gives the sign:

Counterclockwise torques are positiveClockwise torques are negative

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Torque and EquilibriumFirst Condition of Equilibrium

The net external force must be zero

This is a statement of translational equilibrium (we know it from previous chapters)This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium: the object may rotate!

0

0 0x y

or

and

Σ =

Σ = Σ =

F

F F

r

r r

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Torque and Equilibrium, cont

To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translationalThe Second Condition of Equilibrium (for rotation) states

The net external torque must be zero

0τΣ =r

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Equilibrium Example (8.3)The woman, mass m, sits on the left end of the see-sawThe man, mass M, sits where the see-saw will be balancedApply the Second Condition of Equilibrium and solve for the unknown distance, x

Free body diagram

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Axis of RotationIf the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque

The location of the axis of rotation is completely arbitraryOften the nature of the problem will suggest a convenient location for the axisWhen solving a problem, you must specify an axis of rotation

Once you have chosen an axis, you must maintain that choice consistently throughout the problem

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Fig 8.12, p.228

Slide 17

Example (8.7 page 235) Beam

The free body diagram includes the directions of the forcesThe weights act through the centers of gravity of their objects

18Fig. P8-20, p.256

19Fig. 8-2, p.228

20Fig. P8-58, p.260

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Center of Gravity

The force of gravity acting on an object must be consideredIn finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point, called the center of gravity

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Calculating the Center of Gravity

The object is divided up into a large number of very small particles with small weight Each particle will have a set of coordinates indicating its location (x,y)

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Calculating the Center of Gravity, cont.

We assume the object is free to rotate about its centerThe torque produced by each particle about the axis of rotation is equal to its weight times its lever arm

For example, for a particle 1 with mass m1 τ1 = m1 g x1

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Calculating the Center of Gravity, cont.

We wish to locate the point of application of the single forcewhose magnitude is equal to the total weight of the object, and whose effect on the rotation is the same as all the individual particles.This point is called the center of gravity of the object

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Coordinates of the Center of Gravity

The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles being set equal to the torque produced by the weight of the object

i i i icg cg

i i

m x myx and y

m mΣ Σ

= =Σ Σ

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Center of Gravity (C.G.)of a Uniform Object

The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry.Often, the center of gravity of such an object is the geometric center of the object.Example: for the ring, the c.g. is in the center of the ring

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Experimentally Determining the Center of Gravity

The wrench is hung freely from two different pivotsThe intersection of the lines indicates the center of gravityA rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object’s center of gravity

28p.230

29Fig. Q8-12, p.253

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Notes About Equilibrium

A zero net torque does not mean the absence of rotational motion

An object that rotates at uniform angular velocity can be under the influence of a zero net torque

This is analogous to the translational situation where a zero net force does not mean the object is not in motion

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Solving Equilibrium Problems

Draw a diagram of the systemInclude coordinates and choose a rotation axis

Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object

For systems containing more than one object, draw a separate free body diagram for each object

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Problem Solving, cont.Apply the Second Condition of Equilibrium

This will yield a single equation, often with one unknown which can be solved immediately

Apply the First Condition of EquilibriumThis will give you two more equations

Solve the resulting simultaneous equations for all of the unknowns

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Example of a Free Body Diagram (Ladder)

The free body diagram shows the normal force and the force of static friction acting on the ladder at the groundThe last diagram shows the lever arms for the forces

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Torque and Angular Acceleration

When a rigid object is subject to a net torque (≠0), it undergoes an angular accelerationThe angular acceleration is directly proportional to the net torque

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Moment of Inertia

The analog for force for rotation is torqueThe mass analog for the rotation is called the moment of inertia, I, of the object

SI units are kg m2

2I mr≡ Σ

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Newton’s Second Law for a Rotating Object

The angular acceleration is directly proportional to the net torque

The relationship is similar to∑F = ma

Newton’s Second Law for translation

Iτ αΣ =

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QQ2

A constant net torque is applied to an object. Which one of the following will not be constant? (a) angular acceleration (b) angular velocity (c) moment of inertia (d) center of gravity

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More About Moment of Inertia

There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object.The moment of inertia also depends upon the location of the axis of rotationAlso the units are different

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Moment of Inertia of a Uniform Ring

Image the hoop is divided into a number of small segments, m1 …These segments are equidistant from the axis

2 2i iI m r MR= Σ =

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Other Moments of Inertia

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QQ3

The two rigid objects shown in Figure have the same mass, radius, and angular speed. If the same braking torque is applied to each, which takes longer to stop? (a) A (b) B (c) more information is needed

42Fig. 8-16, p.238

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Example, Newton’s Second Law for Rotation

Draw free body diagrams of each objectOnly the cylinder is rotating, so apply Στ= I αThe bucket is falling, but not rotating, so apply ΣF = m aRemember that a = α r and solve the resulting equations

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Rotational Kinetic EnergyAn object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω2

Similar= to the formula for a known formula for kinetic energy (for translational motion) Ek = ½mv2

Energy concepts can be useful for simplifying the analysis of rotational motion

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QQ4Two spheres, one hollow and one solid, are rotating with the same angular speed around an axis through their centers. Both spheres have the same mass and radius. Which sphere, if either, has the higher rotational kinetic energy? (a) The hollow sphere. (b) The solid sphere. (c) They both have the same kinetic energy.

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Total Energy of a SystemConservation of Mechanical Energy

Remember, this is for conservative forces, no dissipative (non-conservative) forces such as friction can be presentPotential energies of any other conservative forces could be added (e.g. spring)

( ) ( )t r g i t r g fKE KE PE KE KE PE+ + = + +

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QQ 5 (Modified)Which arrives at the bottom first: (a) a ball (hollow inside) rolling without sliding down a certain incline A, (b) a solid cylinder rolling without sliding down incline A, or (c) a box sliding down a frictionless incline B having the same dimensions as A?

Assume that all objects have the same mass and each object is released from rest at the top of its incline.

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Work-Energy in a Rotating System

In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of EnergyWnc = ΔKEt + ΔKER + ΔPE

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General Problem Solving Hints

The same basic techniques that were used in linear motion can be applied to rotational motion.

Analogies: F becomes , m becomes I and a becomes , v becomes ω and x becomes θ

τα

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Problem Solving Hints for Energy Methods

Choose two points of interestOne where all the necessary information is givenThe other where information is desired

Identify the conservative and non-conservative forces

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Problem Solving Hints for Energy Methods, cont

Write the general equation for the Work-Energy theorem if there are non-conservative forcesAlternatively use Conservation of Energy if there are no non-conservative forcesUse v =rω if necessary to combine translational and rotational termsSolve for the unknown

52Fig. 8-25, p.245

53Fig. P8-36, p.258

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Angular MomentumSimilarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentumAngular momentum is defined as

L = I ω (similar to M=mv)

and (similar to )Lt

τ ΔΣ =

Δ( )f i

net

mv vt t

−Δ= =

Δ Δp

Fr r

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Angular Momentum, cont

Conservation of Angular Momentum states: The angular momentum of a system is conserved (remains constant) when the net external torque acting on the systems is zero.

That is, when

0, i f i i f fL L or I Iτ ω ωΣ = = =

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Conservation of Angular Momentum, Example

With hands and feet drawn closer to the body, the skater’s angular speed increases

L is conserved, I decreases, ωincreases

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QQ (Example 8.15)

A student walks toward the center of the rotating platform. The angular speed of the platform will:(a) increase(b) decrease(c) stay the same

58Fig. 8-30, p.249

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QQ7 “Al Gore quiz”If global warming continues, it’s likely that some ice from the polar ice caps of the Earth will melt and the water will be distributed closer to the Equator. If this occurs, would the length of the day (one revolution) (a) increase, (b) decrease, or (c) remain the same?

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Conservation Rules, Summary

In an isolated system (with no external forces acting on the system, and no internal dissipation of energy), the following quantities are conserved:

Mechanical energyLinear momentumAngular momentum