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°

θ = sr

θ = the angle

r = the radius

s = arc length

360° = 2π rad

We now have a conversionBetween radians and degrees.

θ (rad) = π180°

θ (deg)

Ben, riding on a carousel, Travels through an arc length of

11.5 m. If the carousel has a Radius of 8 m, what is the angular

Displacement? What degree is this?

2.88 rad 165°

Angular speed describes The rate at which a body

Rotates about an axis, usuallyExpressed in rad/s.

ω = ΔθΔt

Ben now spins on a stool, if heTurns clockwise through 10π rad,

During a 10 s interval, what isThe average angular speed

Of his feet?

3.14 rad/s

Angular acceleration is the timeRate of change of angular speed,

Usually expressed in radians Per second per second.

α = ΔωΔt

Now Ben rotates on the stoolWith 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

Rotational

Motion

Linear

Motion

ωf = ωi + αΔt vf = vi + aΔt

Δθ = ωiΔt + ½α(Δt)2 Δx = viΔt + ½a(Δt)2

ωf2 = ωi

2 + 2α(Δθ) vf2 = vi

2 + 2aΔx

Angular Kinematics

The wheel on an upside-down bikeRotates with a constant angular Acceleration of 3.5 rad/s/s. If the Initial angular speed of the wheelIs 2 rad/s, through what angular

Displacement does the wheelRotate in 2 sec?

11 rad

Torque is a quantity that measuresThe 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 directionThe force tends to rotate

An object.

Torque that is clockwise isDefined as negative.

Thus torques that produce a Counterclockwise rotation are

Defined to be positive.

Find the torque produced by a 3 N force applied at an angleOf 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 theFormulas provided in the bookFor the many different shapes.

Shape Formula for moment of inertia

Thin hoop about symmetry of object

MR2

Thin hoop about diameter 1/2 MR2

Point of mass about axis MR2

Disk about axis 1/2 MR2

Rod about axis through center 1/12 ML2

Rod about axis through end 1/3 ML2

Solid sphere about diameter 2/5 MR2

Thin sphere shell about diameter

2/3 MR2

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 angleOf 53°, and a 545N is standing

1.5m from the wall. Find the tensionAnd the force on the beam by the

Wall, R.

Tension = 403 N R = 590N

Newton’s 2nd Law for Rotating objects.

τ = Iα

A catapult propels a 0.150 kgStone. The length of the

Catapult arm is 0.350m. If the Stone leaves the catapult with an

Acceleration of 100 m/s2, whatIs the torque exerted on the stone.

5.25 Nm

The center of mass of an objectIs the point at which all the

Mass of the body can be Considered to be concentrated

When analyzing Translational motion.

Rotational and translationalMotion can be combined.

The moment of inertia is theRotational analog of mass.

Equilibrium requires zero net force,And zero net torque.

A 5.8 kg ladder, 1.8 m long, restsOn 2 saw horses. Sawhorse A is 0.6 m from one end of the ladder

And sawhorse B is 0.15 m from theOther end of the ladder. What force

Does each sawhorse exert on the Ladder?

Fb = 16 N Fa = 41 N

The centrifugal force is anApparent force.

NOT A REAL FORCE!

It is not a real force because thereIs no physical outward push.

The Coriolis Force is also a FAKE FORCE!

This is because it is only apparentTo rotating observer.

This is very apparent on Earth However. It is the reason forThe direction of the winds.