Angular velocity. Angular velocity measures how quickly the object is rotating. Average angular velocity. Instantaneous angular velocity
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1 Angular velocity Angular velocity measures how quickly the object is rotating. Average angular velocity Instantaneous angular velocity
2 Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B.
3 Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed of A is half that of B.
4 Angular Acceleration Angular acceleration measures how rapidly the angular velocity is changing: Average angular acceleration Instantaneous angular acceleration
5 Linear motion definitions versus rotational motion definitions (average values)
6 Linear kinematics vs. rotational kinematics
7 A bicycle travels 150 m along a circular track of radius 15 m. What is the angular displacement in radians of the bicycle from its starting position?
8 A wheel with a 0.10-m radius is rotating at 30 rev/s. It then slows uniformly to 15 rev/s over a 3.0 second interval. What is the angular acceleration of a point on the wheel?
9 q A graph of angle versus time appears like so: Which of the following angular velocity graphs matches the above angle graph? w w w w A. B. C. D.
10 w A graph of angular velocity versus time for a rolling wheel appears like so: Which of the following angle graphs matches the above angular velocity graph? A. B. C. D.
11 Sign of the Angular Acceleration
12 A ladybug sits at the outer edge of a merry-go- round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug s angular speed is 1. half the ladybug s. 2. the same as the ladybug s. 3. twice the ladybug s. 4. impossible to determine
13 Example A high-speed drill rotating CCW takes 2.5 s to speed up to 2400 rpm from rest. A. What is the drill s angular acceleration? B. How many revolutions does it make as it reaches top speed?
14 Interpretation of Torque Torque is due to the component of the force perpendicular to the radial line. rf rf sin
15 A Second Interpretation of Torque r F rf sin
16 Which factor does the torque on an object not depend on? A. The magnitude of the applied force. B. The object s angular velocity. C. The angle at which the force is applied. D. The distance from the axis to the point at which the force is applied.
17 A net torque applied to an object causes A. a linear acceleration of the object. B. the object to rotate at a constant rate. C. the angular velocity of the object to change. D. the moment of inertia of the object to change.
18 The four forces shown have the same strength. Which force would be most effective in opening the door? A. Force F 1 B. Force F 2 C. Force F 3 D. Force F 4 E. Either F 1 or F 3
19 A string is tied to a doorknob 0.80 m from the hinge as shown in the figure. At the instant shown, the force applied to the string is 5.0 N. What is the torque on the door? hinge
20 Newton s Second Law for Rotation I = moment of inertia. Objects with larger moments of inertia are harder to get rotating.
21 Which has a greater rotational inertia about its center of mass? They are equal
22 You have two uniform solid cylinders of the same material. Cylinder 2 has twice the radius of cylinder 1 but the same length. By what factor does the CM moment of inertia (about symmetry axis) I 2 of cylinder 2 exceed the moment of inertia I 1 of cylinder 1. (1) 4 (2) 8 (3) 16 (4) their moments of inertia are the same.
23 Moments of Inertia of Common Shapes
24 Rolling Is a Combination of Translation and Rotation
25 Rotational Kinetic Energy The kinetic energy of rotating point mass, m 1, is given by For several points rotating, the total kinetic energy is the sum of the kinetic energy of all point masses We know that all points in a rigid body rotate at the same angular velocity, w i = w j = w, where i = 1, 2,...,N and j = 1, 2,...,N. Therefore,
26 Rotational Kinetic Energy (continued) Recall that the moment of inertia was given by the expression, We can now substitute in the expression for the moment of inertia of the object into our rotational kinetic energy equation to get, The total kinetic energy of an object is the sum of the translational kinetic energy of the object s center-of-mass and the rotational kinetic energy about the object s center-of-mass.
27 Angular Momentum Similar to the linear momentum of a moving object, a rotating object has angular momentum, For a closed system, angular momentum is conserved, Using the above expressions, we may rewrite the conservation of momentum equation as
28 A student sits on a rotating stool holding two weights, each of mass 5.00 kg and approximated as point masses. When his arms are extended horizontally, the weights are m from the axis of rotation and the student rotates with an angular speed of 2.00 rad/s. The moment of inertia of the student plus stool is a constant, 3.00 kg m 2. While spinning, the student pulls the weights inward horizontally to a position m from the axis of rotation. Calculate the angular speed of the student after the weights have been pulled inward.
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