Class 41: Torque and Rotational Motion Course Evaluation: 1. Started - PowerPoint PPT Presentation

Class 41: Torque and Rotational Motion

Course Evaluation: 1. Started yesterday Apr 13 th , ends Apr 29 th (Wednesday). 2. Go to http://pa.as.uky.edu/ 3. Click at “UNDERGRADUATES” in the top menu and then choose the first item: Physics & Astronomy Course Evaluations 4. Follow instructions from there. 5. Make sure remember or write down any given key or password. You need this to re-enter the system if you cannot finish the evaluation in one time.

Moment of Inertia of a single particle   M R M “Easy” to change angular velocity “Difficult” to change angular velocity Measured by moment of inertia (direct translation – “angular mass”): I = MR 2

Moment of Inertia Moment of inertia to rotational motion is like mass to translational motion. It measures the difficulty or easiness (inertia) in changing the angular velocity. The moment of inertia not only depends on mass (M), but it also depends on the mass distribution about the axis of rotation. So it depends on the shape of the object and the location and orientation of the axis of rotation.   2 I r dm dm whole r body Unit of moment of inertia: kgm 2

Moment of Inertia of Some Common Shapes For the same M and R, the maximum possible moment of inertia is MR 2 , when all masses are furthest away from the axis if rotation – a ring (or hoop).

Parallel Axis Theorem d X   2 I I Md X CM

cross product between two vectors    A B Direction:  B   Magnitude: A        A B | A || B | sin ˆ ˆ ˆ i j k   A A A A A A    ˆ  ˆ  ˆ y z x y x z A B A A A i j k x y z B B B B B B y z x y x z B B B x y z a b  ad - bc c d

F    r r F Examples F    r r F

Torque       r F r is the displacement vector from the origin to the position of the point at where the force is acting on the rigid body. F 1 F 2       r F r 2 2 2 o M F 3 Unit of torque: Nm

Example Calculate the torque of each of the following forces (the axis of rotation is perpendicular to the page, located at O): 1. 3m 5N 2N 45 o O 1.5m 3N 3kg O 30 o 0.75m 1.5m 2kg

Rotational Motion – 1D In this class we will study only “simple” rotational motion: the axis of rotation is perpendicular to the page. In this simple case, all vectors related to rotational motion like angular velocity  , angular acceleration  , torque  , angular momentum L etc. always along the axis of rotation, perpendicular to the page. Note that their magnitude can be changing with time and they can be either out of the page or into the page (i.e. have sign) – you can call this 1D rotational motion!

Rotational Motion – 1D Warning: Some equations in the “dictionary” I give in the next few slides are valid only for 1D rotational motion. Proper modifications and extension have to be made for more complicated rotational motion when the rotational vectors are not along the same straight line.

Translational and rotation motion formula conversion dictionary – Kinematics Constant angular acceleration Translational Motion Rotational Motion  x  v  a        v v a t t f i f i 1 1           2 2 x x v t a t t t f i i f i i 2 2            2 2 2 2 2 ( ) v v 2a (x x ) f i f i f i f i

Translational and rotation motion formula conversion dictionary – Dynamics Translational Motion Rotational Motion Mass M Moment of inertia I Torque  Force F Newton’s second law  =I  Newton’s second law F=ma Angular momentum L = I  Momentum p = mv 1 1    2 2 Rotational K. E. I K. E. Mv 2 2                        W F d F(x x ) ( F // d ) W ( ) ( // ) f i f i