Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed. Copywrited by Holt, Rinehart, & Winston

Torque ( ) Torque is the product of a force and the perpendicular distance from the axis of rotation to the line of action of the force (aka the lever arm). when F is not perpendicular to a line from the axis of rotation or where F can be written as Fsinθ Copywrited by Holt, Rinehart, & Winston Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Torque ( ) Units: Newton meter (Nm) Positive: Counter Clockwise (ccw) Rotation Max Torque when Force is 90° to the line of action. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed. Copywrited by Holt, Rinehart, & Winston

Equilibrium A rigid body is considered in equilibrium is it has zero translational acceleration and zero angular acceleration. OR 2 conditions required for equilibrium If object has forces acting in both the horizontal and vertical directions then: Copyright ©2007 Pearson Prentice Hall, Inc.

Torque Summation In summing torques since the net torque is the same anywhere on the body, it does not matter which point of rotation we choose for our summation of torques Best to choose a point that is easiest to solve (The location of one of our unknown forces) Copyright ©2007 Pearson Prentice Hall, Inc.

Diver A diver weighing 530N is at the end of a diving board with a length of 3.90m. The board has a negligible weight and is bolted at one end with a fulcrum supporting it at 1.40 m from the end. Find the force of the bolt and the fulcrum on the board. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Moment of Inertia Alpha is the same for all particles of the object and when removed, we get the MOMENT OF INERTIA. This is a property of the object just like Inertia (mass). Different shapes have different values of I. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Moment of Inertia Experiments show that I is directly proportional to the mass. The distribution of mass in the body. To illustrate this consider two wheels having equal mass but different mass distribution. AB

If both of these wheels accelerate from rest to the same angular velocity ω in the same time t. The angular acceleration, α, must be the same for both wheels. Also, the total angle turned through must be the same. But, when moving with angular velocity ω, the particles of wheel B are moving faster than the particles of wheel A. Therefore, B possesses more kinetic energy than wheel A. More work is done accelerating wheel B than wheel A. A greater torque was needed to accelerate B than A. and so the moment of inertia of wheel B is greater than the moment of inertia of wheel A. AB Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Moment of Inertia The moment of inertia of a body is directly proportional to its mass and increases as the mass is moved further from the axis of rotation. The fact that I depends on mass distribution means that the same body can have different moments of inertia depending on which axis of rotation we consider. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Moment of Inertia Proof The moment of inertia of any body can be found by adding together the moments of inertia of all its component particles. Using this idea gives the following results: Copywrited by Holt, Rinehart, & Winston

Center of Gravity Point at which a body’s weight can be considered to act when calculating the torque due to weight. X cg =location of the center of gravity Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed. A synonymous concept is the center of mass and sometimes these terms are interchanged.

Center of Gravity If an extended object is to be balanced, it must be supported through its center of gravity. Copyright ©2007 Pearson Prentice Hall, Inc.

Copywrited by Holt, Rinehart, & Winston Center of Gravity The center of gravity can be physically found by suspending an object by a point and tracing the vertical axis it hangs along. Hang the object from a different point and again tracing its vertical axis. Where these two lines cross would be the center of gravity. Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed. Copyright ©2007 Pearson Prentice Hall, Inc.

CG of an Arm Calculate the center of gravity of the following arm. W 1 =17 N W 2 =11 N W 3 =4.2 N Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Comparing Linear and Rotational Dynamics We can derive new equations by simply substituting rotational quantities for linear quantities. For example: Power is the time-rate of doing work. p = mv gives us What is the last quantity derived called? Linear quantity Angular (rotational) quantity d  vovo 11 vfvf 22 a  F T (Torque) m I (Moment of Inertia) p L L = Iω Angular momentum

Angular Momementum (L) equivalent to linear momentum A "principle of conservation of angular momentum" also exists. L = Iω

Conservation of Angular Momentum With arms and leg outstretched (A), the moment of inertia, I initial, is relatively large. Suppose the skater has a low initial angular velocity ω initial. The skater then gradually decreases her/his moment of inertia by bringing arms and leg nearer to the axis of rotation (B and C). Her/his angular velocity is observed to increase. This is easily explained if we consider that the person’s angular momentum does not change. If I initial > I final then ω final > ω initial I initial ×  initial = I final ×  final Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Conservation of Angular Momentum

Angular Momentum linear momentum multiplied by its distance from the point considered Notice that we can therefore calculate the angular momentum of a body about any point even if the body is not moving in a circular path. L = pr

A High Diver A diver can change his/her rate of rotation in a similar way. The diver starts out with low angular velocity with body straight and arms outstretched. The distribution of mass is then changed in order to vary the angular velocity.

Cat Cats use the principle of conservation of angular momentum in order to rotate themselves "in mid-air". A cat will always land on its feet, if given enough (but not too much!) height.

Copywrited by Holt, Rinehart, & Winston Comparing Linear and Rotational Dynamics

Rotational Kinetic Energy An Object can experience Translational (linear) motion and/or Rotational motion. Both motions have related kinetic energies. Translational kinetic energy is ½ mv 2 The Rotational Kinetic energy is based upon Translation KE: Cutnell & Johnson, Wiley Publishing, Physics 5 th Ed.

Riddle of the Sphinx Beside you are two balls. One is hollow, the other is solid. Tell me which is which. The balls are of equal weight. –The hollow ball is weighted inside with metal, so they are both exactly the same weight. A plank of wood may be used. How will you avoid being the Sphinx’s lunch?

Pulley with Mass A 10 kg pulley of radius 0.5 m is connected to a 5.0 kg mass through a string as shown. If the mass is let go from rest (a) What is its linear acceleration? (b) What is the angular acceleration of the pulley?