Rotational Mechanics

Center of Gravity

The point located at the object’s average position of weight.

The center of gravity does not always lie within the object itself….

When an object is in motion, its center of mass will follow a smooth line.

Center of Mass and Center of Gravity

• Toss a baseball into the air, and it will follow a smooth parabolic trajectory.

• Toss a baseball bat spinning into the air, and its path is not smooth; its motion is wobbly, and it seems to wobble all over the place.

Locating the Center of GravityMethod 1

• Balancing an object provides a simple method of locating its center of gravity.

• The weight of an entire stick behaves as if it were concentrated at the stick’s center of gravity.

Locating the Center of Gravity, CG

Method Two:

If you suspend any object, its cg will be located along a vertical line drawn from the suspension point.

• For a shape that is irregular, the center of gravity lies directly beneath the point of suspension. If we draw a vertical line through the point of suspension, the center of gravity is somewhere along that line.

• To determine exactly where it lies, we have to suspend the object from some other point and draw a vertical line from that point of suspension.

• Where the two lines intersect is the center of gravity.

If the cg remains in a vertical line with the “support base”, an object will not topple over.

However, if the cg is out beyond the support base, it will topple.

Why do pregnant women get backaches?

Pre-AP: Finding the coordinate for the center of mass for 2 separate

masses.

21

2211

mm

xmxmxcm

When there’s no net external forces acting on the masses, if the center of mass was at rest, it will remain at rest, if it was in motion, it continues that motion.

x1

x2

m1 m2

21

2211

mm

xmxmxcm

Get out your map of Texas to turn in.

Put your name on the back.

• Consider the force required to open a door. Is it easier to open the door by pushing/pulling away from hinge (by the door knob) or close to hinge?

closeclose to hinge to hinge

awayaway from from hingehinge

Farther Farther from from from from hinge= hinge= larger larger rotational rotational effect!effect!

Torque

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

– is the torque– d is the lever arm – F is the force

Fd

Door example:

Unit: Newton x meter, Nm

In the illustration below, more torque is applied in the middle example than in the first example because the Force is more effective when it is perpendicular.

In the third example a piece of pipe is used to extend the lever arm providing even more torque.

ConcepTest

You are using a wrench and trying to loosen a rusty nut.

Which of the arrangements shown is most effective in loosening the nut?

List in order of descending efficiency.

Answer: 2, 1 & 4, 3

Pre-AP Lever ArmIn this case, the lever arm is NOT L.

The lever arm is d, the shortest perpendicular distance from the axis of rotation (the nut) to a line drawn along the direction of the force

Lever arm = d = L sin ΦTorque = Force x lever arm

Torque = FLsin Φ

The lever arm is not necessarily the distance between the axis of rotation and point where the force is applied

An Alternative Look at Torque

• The force could also be resolved into its x- and y-components– The x-component, F cos Φ,

produces 0 torque– The y-component, F sin Φ,

does produce torque:

sinFLF is the forceF is the force

L is the distance along the objectL is the distance along the object

Φ is the angle between force and objectΦ is the angle between force and object

LL

What if two or more different forces act along a lever arm?

Net Torque

• The net torque is the sum of all the torques produced by all the forces

Force = weightForce = weight

Net Torque = Weight 1 x lever arm 1 – Weight 2 x lever arm 2

lever armlever arm

Torque = Force x lever arm

200 N

3 meters

Where would the 300 N boy have to sit relative to fulcrum for balanced

torque?

Balanced Torque occurs when the net torque is ZEROThe torque on one side of the fulcrum = the torque on the other side

300 N

??

200 x 3 = 300 x ??

• Notice how the torque is the same for the boy as it is for the girl.

• Even if the girl is suspended by a rope 4 feet below where she was the torques are still the same!

• The lever arm is the perpendicular distance from the pivot point to the line along which the force acts.

weight

QUESTION• A uniform meterstick supported at the 25-

cm mark balances when a 2-kg rock is suspended at the 0-cm end. What is the mass of the meterstick?

What is the location where all the weightof the meter stick is acting?

Weight of meter stick

25 cm 25 cm

2 kg ?

ANSWER

The mass of the meterstick is 2 kg.

The system is in equilibrium, so any torques must be balanced.

Review• Torque is the rotational counterpart of force.• Force tends to change the motion of things; • Torque tends to twist or change the state of

rotating things.

For torque, the perpendicular distance from the axis of rotation is called the lever arm.

Torque is the product of the lever arm and the force that tends to produce rotation:

Torque = lever arm x Force

Angular Momentum

Rotational Inertia

Newton’s 1st Law, the Law of Inertia “An object in motion tends to remain in

motion, unless a net force acts on it.” Its tendency to either remain at rest or in

motion is called its inertia and is measured by taking its mass.

In a similar way, “An object in rotation tends to keep rotating” The resistance of an object to changes in its

rotation is its rotational inertia.

Rotational inertia, I , depends on mass and how that mass is distributed around the axis of rotation. The greater the distance between the bulk of the mass and the axis of rotation, the greater the rotational inertia.

Hoop: the bulk of the mass is as far away from the axis as it could possibly be!

Maximum rotational inertia!

This means this shape is the hardest to start rotating AND the hardest to stop rotating.

Rotational inertia, I , depends on mass and how that mass is distributed around the axis of rotation. The greater the distance between the bulk of the mass and the axis of rotation, the greater the rotational inertia.

Spherical shell: the bulk of the mass is as far away from the axis as it could possibly be!

Lots of rotational inertia (but not as much as a hoop)!

This means this shape is the hard to start rotating AND the hard to stop rotating.

Rotational inertia, I , depends on mass and how that mass is distributed around the axis of rotation. The greater the distance between the bulk of the mass and the axis of rotation, the greater the rotational inertia.

cylinder: the bulk of the mass is evenly distributed

Not as much rotational inertia

This means this shape is the easier to start rotating AND the easier to stop rotating.

Rotational inertia, I , depends on mass and how that mass is distributed around the axis of rotation. The greater the distance between the bulk of the mass and the axis of rotation, the greater the rotational inertia.

Solid sphere: the mass is as closely packed as possible!

Least rotational inertia!

This means this shape is the easiest to start rotating AND the easiest to stop rotating.

A person can change their own rotational inertia by extending your arms, rolling up in a ball, etc.

By doing that, you will rotate or spin more easily or less easily, depending on if you increased or decreased your rotational inertia.

Gymnasts, ice-skaters, divers, and dancers all use this principle.

Objects that are rotating have more STABILITY!

Think of frisbees and bicycles.

Angular Momentum

Object in rotational motion have rotational or angular momentum, L, found by multiplying

rotational inertia x angular velocity

I - rotational inertia

(omega)- angular (rotational) velocity measured in radians per second.

Angular Momentum, L = I

Since angular momentum, I remains constant, if your rotational inertia, I, goes down…

Your velocity, goes up!

But the product of inertia and velocity (original and final) stays the same!

Ioo = Iff

Examples:• Kids on a merry-go-round• Ice-skater• Diver• Student on a spinning chair

Example: An ice skater with a rotational inertia of 80 kg m2 is spinning with velocity of 2 rad/s. When she pulls her arms in, her rotational inertia drops to 5 kg m2. What is her angular momentum? What is her final velocity?

Io= 80 o = 2 Iff = ?

Angular momentum, L = Ioo = 80 x 2 = 160Original L = Final L

Ioo = Iff

f= 80(2) / 5

f= 32 rad / s

Pre-AP Don’t forget….

v = r

= v/r