3. Rotational Motion - refers to motion of a body about a fixed axis of rotation wherein, the particles have the same instantaneous angular velocity.

5. INTRODUCTION Rotational inertia, which is also termed as moment of inertia, is a similar concept applied to objects whose motion is rotational instead of linear. The rotational inertia of an object is that object’s resistance to a change in its angular velocity. That is, if an object is spinning, it has a tendency to keep spinning until a net torque acts upon it.

6. FORMULA For the very special case of the moment of inertia of a single mass m, rotating about an axis, a distance r from m, we have I=mr 2

7. FigureAxisFormula Thin ringAxis through centerI=mr 2 DiskAxis through centerI=½ mr 2 Thin rodAxis through one end perpendicular to length I=1/3 ml 2

8. Sample Problem Find the moment of inertia of a solid cylinder of mass 3.0 kg and radius 0.50 m, which is free to rotate about an axis through its center. Given: m=3.0 kg r=0.50 m Find:I

9. Solution: I = ½ mr 2 = ½ (3.0 kg) (0.50 m) 2 = ½ (3.0 kg) (0.25m 2 ) = 0.38 kg m 2

10. Laws for Rotational Motion 1 st law for rotational motion: A body in motion at a constant angular velocity will continue in motion at that same angular velocity, unless acted upon by some unbalanced external torque.

11. 2 nd law for rotational motion: When an unbalanced external torque acts on a body with moment of inertia I, it gives that body an angular acceleration a, which is directly proportional to the torque T and inversely proportional to the moment of inertia.

12. 3rd law for rotational motion: If body A and body B have the same axis of rotation, and if body A exerts a torque on body B, then body B exerts an equal but opposite torque on body A.

14. Torque is the quantity that measures how effectively a force (F) causes acceleration. a train wheel The magnitude of the torque can be calculated by: T= F x l

15. Sample Problem The width of a door is 40 cm. If it is opened by applying a force of 2 N at its edge (away from the hinges), calculate the torque produced which causes the door to open. Solution : Length of lever arm = 40 cm = 0.40 m (since distance between axis of rotation and line of action of force is 40 cm) Force applied = 2 N Torque = Lever arm x force applied = 0.40 x 20 = 8 Nm (8 Newton meter)

17. The center of gravity is the point from where all the weight of an object is said to be concentrated.

18. Locating the Center of Gravity A system is provided for determining the center of gravity of a body by providing a balance arm supported from a single pivot point, then positioning the body on the balance arm and measuring the rotation of the balance arm in two orthogonal planes.

22. We have just seen that a body in uniform circular motion experiences a centripetal acceleration. From Newton’s second law of motion, there must be a force that gives an object this acceleration. Applying Newton’s Second Law to an object in uniform circular motion, F = ma F c = ma c = mv 2 r F c = mv 2 r

24. Sample Problems A 400-g rock attached to a 1.0 m string is whirled in a horizontal circle at the constant speed of 10.0 m/s. Neglecting the effects of gravity, what is the centripetal force acting on the rock? Given: v = 10.0 m/s r = 1.0 m m = 400 g x 1kg = 0.4 kg 1000 g Find: F c

25. Solution: F c = mv 2 r = (0.4 kg) (100 m 2 /s 2 ) m = 40 kg ∙ m/s 2 = 40 N

27. An object traveling in a circle behaves as if it is experiencing an outward force. This force known as the centrifugal force. F c = mv 2 /r, where F c = centrifugal force, m = mass, v = speed, and r = radius.

29. Simulated gravity is the varying (increase or decrease) of apparent gravity (g- force) via artificial means, particularly in space, but also on the Earth.g- force

31. An angular momentum is the property characterizing the rotary inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system. Angular momentum is a vector quantity, requiring the specification of both a magnitude and a direction for its complete description.rotary inertiamotionmomentum H = I ω H = angular momentum I = inertia w = inertial space

32. Conservation of Angular Momentum Momentum is the product of inertia and velocity. Inertia means the tendency of something not to change, and velocity means how fast it moves. So momentum means the tendency of an object in motion not to slow down. Momentum is of two kinds, angular and linear. Both kinds of momentum are conserved in any collision. Conservation means that none is lost, so the total momentum of the system before the collision, plus any additional impulse from outside the system, will equal the total momentum after the collision. This conservation principle is key to deriving formulas.