Torque changing rotational motion § 10.1–10.2

Example Problem 9.98 A 3.0-kg box is attached by a massless cord over a pulley of mass 2 kg and radius 20 cm and a solid cylinder of mass 5 kg and radius 40 cm. If the box is released from rest, find its speed after it has fallen 1.5 m.

Example Problem If the pulley has no mass: a.What is the machine’s change in potential energy from its initial to its final states? b.With what speed will the heavier mass hit the ground? m1m1 m2m2 h v 0 = 0 mpmp r

Example Problem If the pulley has mass m p and radius r: c.What is the pulley’s kinetic energy when its tangential speed is v? d.What is the kinetic energy of the two masses traveling at speed v? e.With what speed will the heavier mass hit the ground? m1m1 m2m2 mpmp h r

You push on a door. It will open easiest if you push A.opposite the hinge. B.at the center of the door. C.near the hinge. Poll Question

Torque An influence causing angular acceleration angular analogue of Newton’s second law:  net = I –  = torque – I = moment of inertia –  = angular acceleration units?

Torque in Tangential Terms  net = I – For a point-mass, I = mr 2 –  = a tan /r – a tan = F tan /m so  = mr 2 F tan /mr = rF tan r = distance from axis F tan = tangential component of force

Torque Definition  = Fl –F = tangential component of force –l = lever arm or moment arm units?

Lever Arm Shortest distance from line of action to point of interest Source: Young and Freedman, Figure 10.2

Poll Question Force P is applied to one end of a lever of length L. What is the magnitude of the torque about point A ? A. PL sin . B. PL cos . C. PL tan . D. PL sec . E. PL cot . F. PL csc . Source: Young and Freedman, Test Your Understanding §10.1

Torque Vector Turning influence = torque = radius  force = r  F Units: Nm (not J) Source: Hewitt, Conceptual Physics

Vector Cross Product Operation symbol  Another way to multiply two vectors Product is a vector! Direction of A  B is perpendicular to both A and B

Cross Product Magnitude  A  B  = AB sin  A B A B  Maximum for  = 90° Zero for  = 0°, 180°

Reconcile  = PL cos  = PL sin (90° –  ) Source: Young and Freedman, Test Your Understanding § ° – 

Magnitude Geometrically A B A B   A  B  = area of parallelogram

Cross Product Direction Curl right-hand fingers in direction of  Right-hand thumb points in direction of cross- product Not commutative A B A B  AB = –(BA)AB = –(BA)

Poll Question What is the direction of the torque about point O from force F 1 ? Source: Young and Freedman, Figure 10.2 A.  B.  C.  D.  E. F.

Poll Question What is the direction of the torque about point O from force F 2 ? Source: Young and Freedman, Figure 10.2 A.  B.  C.  D.  E. F.

F Point of Interest Can define the torque about any point a point on the rotation axis the center of mass the origin an observer Same force, same line of action, different “axes”

Adding Torques Net torque is zero Source: Hewitt, Conceptual Physics

Whiteboard Work A 10,000-N truck is stalled 1/4 of the way across a 100-m bridge. a.What torque does its weight apply about the far support? r

Whiteboard Work b.What upward force must the near support provide to cancel the truck’s torque about the far support? r F

Whiteboard Work c.What upward force must the far support provide to support the weight of the truck? F r Hint: Several ways will work: forces on the bridge torques about the near support

Poll Question A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated? A.Clockwise. B.Counterclockwise. r

Poll Question A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated? A.Clockwise. B.Counterclockwise. r

Poll Question A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated? A.Clockwise. B.Counterclockwise. r