Concept Summary Belton High School Physics

Circular Motion Terms The point or line that is the center of the circle is the axis of rotation.The point or line that is the center of the circle is the axis of rotation. If the axis of rotation is inside the object, the object is rotating (spinning).If the axis of rotation is inside the object, the object is rotating (spinning). If the axis of rotation is outside the object, the object is revolving.If the axis of rotation is outside the object, the object is revolving.

Planar, rigid object rotating about origin O. Rotational variables Look at one point P: Arc length s: Thus, the angular position is:  is measured in degrees or radians (more common)

One radian is the angle subtended by an arc length equal to the radius of the arc.  r s = r For full circle: Full circle has an angle of 2  radians, Thus, one radian is 360°/2  Radian degrees 2  360°  180°  90° 157.3°

2 pod racers complete a 180° turn and remain neck and neck. Which had the greater Linear Velocity? A.The one closer to the point of rotation. B.The one farther from the point of rotation. C.Both were equal.

How would you describe the time it takes for each of the 2 pod racers to complete the turn? A.Greater for the one closer to the point of rotation. B.Greater for the one farther from the point of rotation. C.Both the same.

Angular Velocity rotational or angular velocity, which is the rate angular position changes.rotational or angular velocity, which is the rate angular position changes. Rotational velocity is measured in radians/sec, degrees/second, rotations/minute (rpm), etc.Rotational velocity is measured in radians/sec, degrees/second, rotations/minute (rpm), etc. Common symbol,  (Greek letter omega)Common symbol,  (Greek letter omega) 

Define quantities for circular motion (note analogies to linear motion!!) Angular displacement: Average angular speed: Instantaneous angular speed: Average angular acceleration: Instantaneous angular acceleration:

10-5 Relating the linear and angular variables Tangential speed of a point P: Tangential acceleration of a point P: Note, this is not the centripetal acceleration a r !! Arc length s: Caution: Measure angular quantities in radians

Rotational Vs. Tangential Velocity If an object is rotating: – All points on the object have the same rotational (angular) velocity. –All points on the object do not have the same linear (tangential) velocity

Rotational & Tangential Velocity We now see that…. –Tangential (linear) velocity of a point depends on: The rotational velocity of the point. –More rotational velocity can mean more linear velocity. –The distance from the axis of rotation. More distance from the axis means more linear velocity

Centripetal Acceleration and Angular Velocity The angular velocity and the linear velocity are related (v = ωr)The angular velocity and the linear velocity are related (v = ωr) The centripetal acceleration can also be related to the angular velocityThe centripetal acceleration can also be related to the angular velocity

10-3 Are angular quantities vectors? Right-hand rule for determining the direction of this vector. rotates through the same angle, has the same angular velocity, has the same angular acceleration. Every particle (of a rigid object):  characterize rotational motion of entire object

Linear motion with constant linear acceleration, a Rotational motion with constant rotational acceleration, .

Rotational Inertia Equates to “normal” Inertia (mass). –An object rotating about an axis tends to keep rotating about that axis. Rotational Inertia: resistance to changes in rotational motion.

Rotational inertia (or Moment of Inertia) I of an object depends on: - the axis about which the object is rotated. - the mass of the object. - the distance between the mass(es) and the axis of rotation. - Note that  must be in radian unit. The SI unit for I is kg.m 2 and it is a scalar. Calculation of Rotational inertia

Note that the moments of inertia are different for different axes of rotation (even for the same object)

Torque Every time you open a door, pull a lever, or use a wrench you exert a “turning” force. This turning force produces a Torque. –Forces make things accelerate. –Torques make things rotate –Also known as a “couple” or “moment”

Producing Torques A torque is produced by “leverage” –Greater the “lever” or length of the lever arm, greater the torque. –Greater the force you apply to the lever arm, greater the torque. AND the angle of the applied force plays a part. –Torque is the Cross Product of Force X Lever arm.

The Cross Product A way of multiplying vectors to produce a different vector. –NOT the same as the Dot product (produced a scalar). Example: Torque = r F sin θ

Torque is positive if the direction of rotation is counterclockwise. Torque is negative if the direction of rotation is clockwise. The SI unit of torque is N.m (Note that the unit of work J is also N.m. However, the name J is exclusively reserved for work/energy) Torque

It is clear that torque can also be defined as We use the right-hand rule, sweeping the fingers of the right hand from to. The outstretched right thumb then gives the direction of. When several torques act on a body, the net torque is the sum of the individual torques, taking into account of positive and negative torques. Newton ’ s Second law can be applied to Torques! –An object will rotate in the direction of the net Torque! –If the Net Torque is zero then no rotation occurs!

Checkpoint: Assuming all of these forces are equal in magnitude, which direction does the net Torque point? A: to the right B: to the left C: into the board D: out of the board

If a doorknob were placed in the center of a door rather than at the edge, how much more force would be needed to produce the same torque for opening the door? a)Equal amount b)Twice as much c)Four times as much d)None of the above

Suppose that a meterstick is supported at the center, and a 20 N block is hung at the 80 cm mark. Another block of unknown weight just balances the system when it is hung at the 10 cm mark. What is the weight of the second block? a)5 N b)10 N c)15 N d)20 N

Homework (by Wednesday) Watch Walter Lewin lecture 19: Rotating Rigid BodiesWatch Walter Lewin lecture 19: Rotating Rigid Bodies And lecture 20: Angular momentum and TorquesAnd lecture 20: Angular momentum and Torques –Available on iTunesU or youtube or my site.