Chapter 11 Rotational Dynamics and

Static Equilibrium

Chapter 11: Rotational Dynamics and Static Equilibrium

Torque: The ability of a force to rotate a body

about some axis. rF Note: F r

The torque is larger if the force is applied farther from the axis of rotation.

By convention, the sign of torque is:

is negative clockwise (cw)

is positive counter-clockwise (ccw)

General Definition of TorqueOnly the component of the force that is perpendicular to the radius causes a torque.

= r(Fsin)

Equivalently, only the perpendicular distance between the line of force and the axis of rotation, known as the moment arm r, can be used to calculate the torque.

= rF = (rsin)F

F

Fsin

Each force that acts on an object may cause a torque.F1

When discussing torques, we must identify a pivot point (or axis of rotation).

In this figure, the three forces have equal magnitude.

• Which forces cause a torque?

• Which force causes the biggest magnitude torque?

• Which forces, if any, causes a positive torque?

r1r2

F2

F3

pivot point

The net torque about a point O is the sum of all torques about O:

A person holds a 1.42 N baseball in his hand, a distance of 2L = 34 cm from the elbow joint, as shown in the figure. The biceps, attached at a distance of d = 2.75 cm from the elbow, exert an upward force of 12.8 N on the forearm. Consider the forearm and hand to be a uniform rod with a mass of 1.39 kg.

(a) Calculate the magnitude of the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. [2.44 N.m]

(b) If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate?[clockwise]

HW 11 problem # 1

Moment of InertiaRecall that mass (inertia) is an object’s resistance to

acceleration. Similarly an object’s resistance to rotation (angular acceleration) is known as moment of inertia. For a point mass m:

I = mr2

I = moment of inertiar = distance from the axis of rotationFor an extended object:

I =miri2

Mass near the axis of rotation resists rotation less than mass far from the axis of rotation.

Solid Sphere

                     

Spherical Shell

                     

Hoop orCylindrical Shell

                     

Solid Cylinderor Thin Disk

                       

Thin Rod or Bar

                                 

Thin Rod about its end

                                 

For circular motion, the distance (arc length) s, the radius r, and the angle are related by:

r

s

DEGRAD 180

Note that is measured in radians:

Angular Position,

> 0 for counterclockwise rotation from reference line

1 rev = 360° = 2 rad

Consider a rotating disk:

t = 0

O P

r

t > 0

O

Pr s

Notice that as the disk rotates, changes. We define the angular displacement, , as:

= f - i

which leads to the average angular speed av

if

ifav ttt

Angular Velocity,

PeriodThe period of rotation is the time it takes to complete one revolution.

T

2 T = period

Rearranging we have2

T

What is the period of the Earth’s rotation about its own axis?

What is the angular velocity of the Earth’s rotation about its own axis?

We can also define the average angular acceleration av:

and

if

ifav ttt

tt

lim0

Angular Acceleration,

The SI units of are: rad/s2 = s-2

We will skip any detailed discussion of angular acceleration, except to note that angular acceleration is the time rate of change of angular velocity

Torque and Angular Acceleration

Recall Newton’s Second Law: F = ma

The net force on an object of mass m causes a (linear) acceleration a.

Similarly, the net torque on an object with moment of inertia I causes an angular acceleration .

= I

HW11 - Problem

When a ceiling fan rotating with an angular speed of 2.15 rad/s is turned off, a frictional torque of 0.241 N m slows it to a stop in 6.25 s. What is the moment of inertia of the

fan? [0.701] kg m2

Consider the wheel shown below. Two forces of equal magnitude are acting on the wheel. Will the wheel remain at rest?

The net force is zero, so there will be no linear acceleration.

Zero Torque and Static Equilibrium

However, the sum of the torques is not zero, so there will be an angular acceleration.

The wheel is not in static equilibrium.

F1

F2

Conditions for Static Equilibrium

For true static equilibrium, two conditions must be satisfied:

For an object in equilibrium, the axis of rotation is arbitrary (But all torques must be evaluated about a common axis).

0

0

F 0

0

y

x

F

F

Angular Momentum

For linear momentum:

p = mv

For rotational motion, we define an angular momentum:

L = I

The SI units of angular momentum are kg·m2/s

Angular Momentum - Problem

A 0.013 kg record with a radius of 15 cm rotates with an angular speed of 29 rpm. Find the angular momentum of

the record. [4.44E-4] kg m2/s

Kinetic energy of rotation

What is the kinetic energy of a mass m traveling at speed v in a circle of radius r?K = (1/2) m v2 = (1/2) mr2 (v/r) 2 = (1/2) I 2

Kinetic energy of rotation = (1/2) I 2

This is not a new form of energy, just a re-labeling (or alternate formula) for kinetic energy.

Calculate the rotational kinetic energy of the Earth as it (a) orbits the sun (b) rotates about its axis.

Mass of Earth = 5.98E24 kgRadius of Earth (ave) = 6.38E6 m

Average Earth-Sun distance = 1.50E11 m

Rotational Kinetic Energy - Problem