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Chapter 8 Rotational Motion

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Chapter 8

Rotational Motion

8.1 Describing Rotational Motion

When an object spins,

It is said to undergo

Rotational motion.

When measuring rotational

Motion, we will use radians.

A radian is an angle whose

Arc length is equal to its radius,

Which is approximately

Equal to 57.3°

s

r

θ =

θ = the angle

r = the radius

s = arc length

360° = 2π rad

We now have a conversion

Between radians and degrees.

π

180°

θ (rad) =

θ (deg)

Problem...

Ben, riding on a carousel,

Travels through an arc length of11.5 m. If the carousel has a

Radius of 8 m, what is the angular

Displacement? What degree is this?

2.88 rad165°

Angular speed describes

The rate at which a body

Rotates about an axis, usually

Expressed in rad/s.

Δθ

Δt

ω =

Problem...

Ben now spins on a stool, if he

Turns clockwise through 10π rad,

During a 10 s interval, what is

The average angular speed

Of his feet?

3.14 rad/s

Angular acceleration is the time

Rate of change of angular speed,

Usually expressed in radians

Per second per second.

Δω

Δt

α =

Problem...

Now Ben rotates on the stool

With an initial angular speed of

21.5 rad/s. The stool accelerates,

And after 3.5s, the speed is 28

rad/s. What is the average

Angular acceleration?

1.9 rad/s/s

Angular Kinematics

Problem...

The wheel on an upside-down bike

Rotates with a constant angular

Acceleration of 3.5 rad/s/s. If the

Initial angular speed of the wheel

Is 2 rad/s, through what angular

Displacement does the wheel

Rotate in 2 sec?

11 rad

8.2 Rotational Dynamics

Torque is a quantity that measures

The ability of a force to rotate

An object around some axis

Torque is a scalar quantity.

Torque has the SI unit of

N*m

τ = Fd(sinθ)

Torque is positive or negative

Depending on the direction

The force tends to rotate

An object.

Torque that is clockwise is

Defined as negative.

Thus torques that produce a

Counterclockwise rotation are

Defined to be positive.

Problem...

Find the torque produced by a

3 N force applied at an angle

Of 60° to a door 0.25m from

The hinge.

0.65 N m

In order to calculate the moment

Of inertia, you have to use the

Formulas provided in the book

For the many different shapes.

Problem...

A 5m horizontal beam weighing

315N is attached to a wall so

That it rotates. Its far end is

Supported by a cable at an angle

Of 53°, and a 545N is standing

1.5m from the wall. Find the tension

And the force on the beam by the

Wall, R.

Tension = 403NR = 590N

Newton’s 2nd Law for

Rotating objects.

τ = Iα

Problem...

A catapult propels a 0.150 kg

Stone. The length of the

Catapult arm is 0.350m. If the

Stone leaves the catapult with an

Acceleration of 100 m/s2, what

Is the torque exerted on the stone.

5.25 Nm

8.3 Equilibrium

The center of mass of an object

Is the point at which all the

Mass of the body can be

Considered to be concentrated

When analyzing

Translational motion.

Rotational and translational

Motion can be combined.

The moment of inertia is the

Rotational analog of mass.

Equilibrium requires zero net force,

And zero net torque.

Problem...

A 5.8 kg ladder, 1.8 m long, rests

On 2 saw horses. Sawhorse A is

0.6 m from one end of the ladder

And sawhorse B is 0.15 m from the

Other end of the ladder. What force

Does each sawhorse exert on the

Ladder?

Fb = 16 NFa = 41 N

The centrifugal force is an

Apparent force.

NOT A REAL FORCE!

It is not a real force because there

Is no physical outward push.

The Coriolis Force is also a

FAKE FORCE!

This is because it is only apparent

To rotating observer.

This is very apparent on Earth

However. It is the reason for

The direction of the winds.

THE END