Rotational Motion

Center of GravityCenter of Gravity (CG):Center of Gravity (CG): an object’s an object’s

average position of weightaverage position of weight for symmetrical objects the CG is at the geometric

center of the object (ball) the CG of a asymmetrical object is located closer to

the end with the most mass (bat & toy) Ex: the CG of a meterstick is at the 50 cm mark; the

CG of a bat is towards the more massive end

Paths of CG• When objects are thrown, they will spin about

their CG and move as if all the weight was concentrated about their CG. The CG of thrown objects will follow are parabolic

path. Objects whose CG is not at their geometric center will

“wobble” about the CG (bat)

• When you slide an object across a horizontal table, the object will rotate about is CG while traveling in a straight line.

• The white dot is the location of the wrench’s CG. Notice that it travels in a straight line

Locating the Center of Gravity• The CG of a uniform object (such as a meter

stick) is at the midpoint, its geometric center• The CG of objects can be located where there is

no matterExamples: A pot, chair, tea cup, boomerang

Check Your UnderstandingCheck Your UnderstandingWhere is the CG of a donut?Where is the CG of a donut?It’s in the center of the hole

Can an object have more than one CG?Can an object have more than one CG?A rigid object has only one CG. If it is

malleable, like putty or clay or the solar system, and is distorted into different shapes, then its CG may change as its shape is changed. Even then, there is only one CG for any given shape.

Toppling (Falling Over)• Rule for toppling: If the CG of an object is above

its base of support, the object will remain upright. If the CG of an object extends beyond the base of support, the object will topple.

The Leaning Tower of Pisa• The Leaning Tower of Pisa does not topple

because its CG is above its base of support

• The base of support does not need to be solid.

• The four legs of a chair bound a rectangular area that is the base of support for the chair

Check Your UnderstandingCheck Your Understanding

When you carry a heavy load – such as a When you carry a heavy load – such as a pail of water – with one arm, why do pail of water – with one arm, why do you tend to hold your free arm out you tend to hold your free arm out horizontally?horizontally?

To shift your CG back over the support base

To resist being toppled, why does a To resist being toppled, why does a wrestler stand with feet wide apart?wrestler stand with feet wide apart?

To increase the support base

To resist being toppled, why To resist being toppled, why does a wrestler stand with does a wrestler stand with knees bent?knees bent?To lower the CG

Check Your UnderstandingCheck Your Understanding

Stability• An object is considered to be stable

when work must be done to raise it’s CG• Three terms to describe an object’s

stability1. Stable equilibrium2. Unstable equilibrium3. Neutral equilibrium

Stable Equilibrium:Stable Equilibrium: any any displacement raises an object’s center displacement raises an object’s center of gravityof gravity Object is less likely to topple Has a wider base of support Ex: a pyramid

Unstable Equilibrium:Unstable Equilibrium: any any displacement lowers an object’s center displacement lowers an object’s center of gravityof gravity Object is more likely to topple Has a narrower base of support Ex: an ice cream cone

Neutral Equilibrium:Neutral Equilibrium: displacement displacement neither raises nor lowers an object’s neither raises nor lowers an object’s center of gravitycenter of gravity Object cannot topple over without putting work

into it Ex: a fish in water, a bat lying on the ground

• The 1st cone is in unstable equilibrium, meaning it is more likely to topple

• The 2nd cone is in stable equilibrium, meaning it is less likely to topple

• The 3rd cone is in neutral equilibrium, meaning it can’t topple over anymore in that position

CG and Stability

• An object becomes more stable when its CG is below the point of support. Icebergs do not fall over because their CG are

below the surface of the water• The CG of objects tend to take the lowest

position available. This is why sport players tend to squat lower

to the ground, to stabilize themselves against an attack from an opponent.

Difference between torque and force

• If you want to make an object move, apply a force

• If you want to make an object rotate, apply a torque.

• Forces produce acceleration.• Torque produce rotation

TorqueTorque:Torque: the perpendicular force times the perpendicular force times

the lever arm lengththe lever arm length Torque produce rotations The force must be perpendicular to the lever

arm Ex: turning a door knob; bending your arm Equation: ττ = F = F┴┴ll

• τ = Torque (N*m)• F┴ = Force perpendicular (N)• l = lever arm length (m)

• The lever arm length is distance from the fulcrum to the area where the force is perpendicularly applied.

Fulcrum:Fulcrum: the pivot point of a lever the pivot point of a lever Where rotation begins Ex: hinge of a door, center of a seesaw, your

knee or elbow

• Torque and lever arm length are directly proportional. 2x lever arm length = 2 x Torque ½ lever arm length = ½ Torque 3x lever arm length = 3x Torque 1/3 lever arm length = 1/3 Torque

• Torque and force are directly proportional 2x force = 2 x Torque ½ force = ½ Torque 3x force = 3x Torque 1/3 force = 1/3 Torque

One way to produce more torque• Although the magnitudes of the applied forces

are the same, the torques are different. Only the component of forces perpendicular to the lever arm contributes to the force.

Check Your UnderstandingCheck Your UnderstandingIf you cannot exert enough torque to turn a If you cannot exert enough torque to turn a

stubborn bolt, would more torque be stubborn bolt, would more torque be produced if you fastened a length of rope to produced if you fastened a length of rope to the wrench as shown?the wrench as shown?

No, because the lever arm (the arm of the wrench) is the same. If you wanted to increase the lever arm, you would need to use a longer wrench.

Balanced Torques• Weight does not produce rotation, torque does. • Consider a heavy boy and a slim boy on a

seesaw. In order to balance the seesaw, the heavy boy must sit closer to the middle than the slim boy. In this way, the counter clock wise torque produced by the slim boy will equal the clock wise torque produced by the heavy boy.

Check Your UnderstandingCheck Your Understanding

Two children are on a seesaw. Child A is twice as heavy as Child B. Which one will sit closer towards the center to make them balanced?

Child A because a larger mass must have a smaller lever arm length to balance the smaller mass with the larger lever arm length of Child B.