1. TORQUE A torque is an action that causes objects to rotate. Torque is not the same thing as force. For rotational motion, the torque is what is most directly related to the motion, not the force.

2. TORQUE Motion in which an entire object moves is called translation. Motion in which an object spins is called rotation. The point or line about which an object turns is its center of rotation. An object can rotate and translate.

3. TORQUE Torque is created when the line of action of a force does not pass through the center of rotation. The line of action is an imaginary line that follows the direction of a force and passes though its point of application.

4. TORQUE To get the maximum torque, the force should be applied in a direction that creates the greatest lever arm. The lever arm is the perpendicular distance between the line of action of the force and the center of rotation

6. TORQUE t = r x F Lever arm length (m) Force (N) Torque (N. m)

7. CALCULATE A TORQUE A force of 50 newtons is applied to a wrench that is 30 centimeters long.  Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.

8. ROTATIONAL EQUILIBRIUM When an object is in rotational equilibrium, the net torque applied to it is zero. Rotational equilibrium is often used to determine unknown forces. What are the forces (F A, F B ) holding the bridge up at either end?

9. ROTATIONAL EQUILIBRIUM

10. CALCULATE USING EQUILIBRIUM A boy and his cat sit on a seesaw. The cat has a mass of 4 kg and sits 2 m from the center of rotation. If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?

11. WHEN THE FORCE AND LEVER ARM ARE NOT PERPENDICULAR

12. CALCULATE A TORQUE It takes 50 newtons to loosen the bolt when the force is applied perpendicular to the wrench. How much force would it take if the force was applied at a 30-degree angle from perpendicular?  A 20-centimeter wrench is used to loosen a bolt.  The force is applied 0.20 m from the bolt.

13. CENTER OF MASS Key Question: How do objects balance?

14. CENTER OF MASS There are three different axes about which an object will naturally spin. The point at which the three axes intersect is called the center of mass.

15. FINDING THE CENTER OF MASS If an object is irregularly shaped, the center of mass can be found by spinning the object and finding the intersection of the three spin axes. There is not always material at an object’s center of mass.

17. FINDING THE CENTER OF GRAVITY The center of gravity of an irregularly shaped object can be found by suspending it from two or more points. For very tall objects, such as skyscrapers, the acceleration due to gravity may be slightly different at points throughout the object.

18. BALANCE AND CENTER OF MASS For an object to remain upright, its center of gravity must be above its area of support. The area of support includes the entire region surrounded by the actual supports. An object will topple over if its center of mass is not above its area of support.

20. ROTATIONAL INERTIA Key Question: Does mass resist rotation the way it resists acceleration?

21. ROTATIONAL INERTIA Inertia is the name for an object’s resistance to a change in its motion (or lack of motion). Rotational inertia is the term used to describe an object’s resistance to a change in its rotational motion. An object’s rotational inertia depends not only on the total mass, but also on the way mass is distributed.

22. LINEAR AND ANGULAR ACCELERATION a = a r Radius of motion (m) Linear acceleration (m/sec 2 ) Angular acceleration (kg)

23. ROTATIONAL INERTIA To put the equation into rotational motion variables, the force is replaced by the torque about the center of rotation. The linear acceleration is replaced by the angular acceleration.

24. ROTATIONAL INERTIA A rotating mass on a rod can be described with variables from linear or rotational motion.

25. ROTATIONAL INERTIA The product of mass × radius squared (mr 2 ) is the rotational inertia for a point mass where r is measured from the axis of rotation.

26. MOMENT OF INERTIA The sum of mr 2 for all the particles of mass in a solid is called the moment of inertia (I). A solid object contains mass distributed at different distances from the center of rotation. Because rotational inertia depends on the square of the radius, the distribution of mass makes a big difference for solid objects.

27. MOMENT OF INERTIA The moment of inertia of some simple shapes rotated around axes that pass through their centers.

28. ROTATION AND NEWTON'S 2ND LAW If you apply a torque to a wheel, it will spin in the direction of the torque. The greater the torque, the greater the angular acceleration.

29. ANGULAR MOMENTUM Investigation Key Question: How does the first law apply to rotational motion?

30. ANGULAR MOMENTUM Momentum resulting from an object moving in linear motion is called linear momentum. Momentum resulting from the rotation (or spin) of an object is called angular momentum.

31. CONSERVATION OF ANGULAR MOMENTUM Angular momentum is important because it obeys a conservation law, as does linear momentum. The total angular momentum of a closed system stays the same.

32. CALCULATING ANGULAR MOMENTUM Angular momentum is calculated in a similar way to linear momentum, except the mass and velocity are replaced by the moment of inertia and angular velocity. Angular velocity (rad/sec) Angular momentum (kg m/sec 2 ) L = I  Moment of inertia (kg m 2 )

33. CALCULATING ANGULAR MOMENTUM The moment of inertia of an object is the average of mass times radius squared for the whole object. Since the radius is measured from the axis of rotation, the moment of inertia depends on the axis of rotation.

34. 1.You are asked for angular momentum. 2.You are given mass, shape, and angular velocity. Hint: both rotate about y axis. 3.Use L= I , I hoop = mr 2, I bar = 1 / 12 ml 2 CALCULATING ANGULAR MOMENTUM An artist is making a moving metal sculpture. She takes two identical 1 kg metal bars and bends one into a hoop with a radius of 0.16 m. The hoop spins like a wheel. The other bar is left straight with a length of 1 meter. The straight bar spins around its center. Both have an angular velocity of 1 rad/sec. Calculate the angular momentum of each and decide which would be harder to stop.

35. 3.Solve hoop: I hoop = (1 kg) (0.16 m) 2 = 0.026 kg m 2 L hoop = (1 rad/s) (0.026 kg m 2 ) = 0.026 kg m 2 /s 4.Solve bar: I bar = ( 1 / 12 )(1 kg) (1 m) 2 = 0.083 kg m 2 L bar = (1 rad/s) (0.083 kg m 2 ) = 0.083 kg m 2 /s 5.The bar has more than 3x the angular momentum of the hoop, so it is harder to stop. CALCULATING ANGULAR MOMENTUM