Rotational MotionRotational Motion

Chapter 6, 8 and 9Chapter 6, 8 and 9

Acceleration in a CircleAcceleration in a Circle

Acceleration occurs when velocity Acceleration occurs when velocity changes changes

This means either speed OR direction This means either speed OR direction changeschanges

So objects moving in a circle are So objects moving in a circle are accelerating even if speed remains accelerating even if speed remains constant because they are constantly constant because they are constantly changing directionchanging direction

Centripetal AccelerationCentripetal Acceleration In order to accelerate, there In order to accelerate, there

must be a net force in the must be a net force in the direction of acceleration direction of acceleration according to Newton’s 2according to Newton’s 2ndnd LawLaw

This means there must be a This means there must be a center- directed forcecenter- directed force

This is called centripetal forceThis is called centripetal force Without centripetal force, Without centripetal force,

inertia would cause the object inertia would cause the object to continue in a straight line at to continue in a straight line at a constant speeda constant speed

Centrifugal ForceCentrifugal Force When moving quickly in a circle, you When moving quickly in a circle, you

feel like you are being pushed outwardfeel like you are being pushed outward This is called centrifugal forceThis is called centrifugal force

The is no outward force, only a inward The is no outward force, only a inward force (centripetal force)force (centripetal force)

Centrifugal force is an imaginary force Centrifugal force is an imaginary force because it doesn’t have a reaction force to because it doesn’t have a reaction force to accompany itaccompany it

You feel the outward force because You feel the outward force because inertia wants you to keep moving in a inertia wants you to keep moving in a straight line, but the centripetal force straight line, but the centripetal force forces you to move in a circle insteadforces you to move in a circle instead

Angular velocity (ω)Angular velocity (ω)

A measure of what angle an A measure of what angle an object is able to travel per object is able to travel per unit timeunit time Unit is rad/sUnit is rad/s All parts of a rigid body rotate All parts of a rigid body rotate

with the same ω, that means with the same ω, that means object’s near the edge have to object’s near the edge have to cover more distance in the cover more distance in the same amount of time (have a same amount of time (have a higher tangential velocity)higher tangential velocity)

Angular measures differ from Angular measures differ from centripetal measures centripetal measures because the object is rotating because the object is rotating around it’s center of mass around it’s center of mass instead of orbiting an outside instead of orbiting an outside pointpoint

Angular Acceleration (Angular Acceleration (αα))

A measure of how quickly angular A measure of how quickly angular velocity is changingvelocity is changing Unit is rad/sUnit is rad/s22

Again, this differs from centripetal Again, this differs from centripetal acceleration because it is rotation of an acceleration because it is rotation of an object around its center of mass as object around its center of mass as opposed to revolving around an external opposed to revolving around an external pointpoint

Starting RotationStarting Rotation

Caused by torque (Caused by torque (ττ) acting on an ) acting on an objectobject This is rotational forceThis is rotational force Unit is a NmUnit is a Nm

Two parts to torque:Two parts to torque: Lever armLever arm

To get the most effect, effort force To get the most effect, effort force should be exerted as far from the axis should be exerted as far from the axis of rotation as possible (why doorknobs of rotation as possible (why doorknobs are at the edge of a door)are at the edge of a door)

L = r, if the force is exerted L = r, if the force is exerted perpendicular to the axis of rotation perpendicular to the axis of rotation

ForceForce Often the weight of an object (Fw = mg)Often the weight of an object (Fw = mg)

Net TorqueNet Torque

If clockwise torque = counterclockwise If clockwise torque = counterclockwise torque, then net torque is zero and no torque, then net torque is zero and no rotation occursrotation occurs This is called static equilibrium or translational This is called static equilibrium or translational

equilibriumequilibrium There is no velocity or accelerationThere is no velocity or acceleration

Moment of Inertia (I)Moment of Inertia (I)

Not only mass matters for Not only mass matters for rotation, its location also rotation, its location also mattersmatters

The further from the axis a The further from the axis a mass is, the harder it is to turnmass is, the harder it is to turn This is why you choke up on a This is why you choke up on a

baseball bat to make it easier to baseball bat to make it easier to swingswing

Can change this by changing Can change this by changing the mass or where the mass the mass or where the mass is located in relationship to is located in relationship to the axis of rotationthe axis of rotation

Newton’s 2nd Law ModifiedNewton’s 2nd Law Modified

Normally, acceleration is equal to force Normally, acceleration is equal to force divided by mass divided by mass

In rotational motion, force is replaced by In rotational motion, force is replaced by torque and mass is replaced by moment torque and mass is replaced by moment of inertiaof inertia

The same equation, with distance from The same equation, with distance from axis of rotation added to account for axis of rotation added to account for circular motioncircular motion

Center of Mass (COM)Center of Mass (COM) Each object has a center of mass (COM)Each object has a center of mass (COM) This COM follows all motion laws, the rest of the object This COM follows all motion laws, the rest of the object

rotates around this pointrotates around this point To find COM, suspend the object at 2 different points. Draw To find COM, suspend the object at 2 different points. Draw

a vertical line down the object from that point. Where the a vertical line down the object from that point. Where the two lines cross is the COMtwo lines cross is the COM This is typically higher on a male’s body then a female’sThis is typically higher on a male’s body then a female’s

You can change your COM by changing your shapeYou can change your COM by changing your shape COM can be located in empty space (ex. donut)COM can be located in empty space (ex. donut)

TopplingToppling

Objects topple when their COM is no Objects topple when their COM is no longer over its support base (τ net no longer over its support base (τ net no longer = 0)longer = 0)

Considered stable if an external force is Considered stable if an external force is needed to cause topplingneeded to cause toppling

The lower the COM, the more stable the The lower the COM, the more stable the object object

Angular Momentum (L)Angular Momentum (L)

Like linear momentum, but with all our Like linear momentum, but with all our modified angular measuresmodified angular measures

Is the product of momentum of inertia Is the product of momentum of inertia and angular velocityand angular velocity

The product of torque and time is the The product of torque and time is the angular impulse which causes a angular impulse which causes a change in angular momentumchange in angular momentum It’s still conserved, like linear momentumIt’s still conserved, like linear momentum So, if your decrease your radius (and So, if your decrease your radius (and

therefore your I), your angular velocity must therefore your I), your angular velocity must increase increase

This is why you spin faster when you pull in This is why you spin faster when you pull in your arms when ice skatingyour arms when ice skating