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Chapter 8
Rotational Motion
8-1 Angular Quantities
- In purely rotational motion, all points on the object move in circles around the axis of rotation (“ O ”).
- All points on a straight line drawn through the axis move through the same angle in the same time.
- The angle θ in radians is defined: where l is the arc length. (8-1a)
Think-Pair-Share
- Problem 1: Express the following angle in
radians: (a) 30 degrees, (b) 57 degrees.
Give as numerical values and as fractions
of pi.
8-1 Angular Quantities
- Angular displacement:
- Average angular velocity:
- The instantaneous angular velocity: (8-2a) (8-2b)
8-1 Angular Quantities
The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration: (8-3a) (8-3b) Think-Pair-Share
- Problem 4: The blades of a blender rotate
at a rate of 6500 rpm. When the motor is
turned off during operation, the blades
slow to rest in 3.0 s. What is the angular
acceleration as the blades slow down?
8-1 Angular Quantities
Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related: (8-4)
8-1 Angular Quantities
Therefore, objects farther from the axis of rotation will move faster.
8-2 Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones. Think-Pair-Share
- Problem 15: A centrifuge accelerates
uniformly from rest to 15,000 rpm in 220 s.
Through how many revolutions did it turn
in this time?
8-3 Rolling Motion (Without Slipping)
- Rolling without slipping - static friction
- wheel contact point is a rest WRT ground
- The linear speed of the wheel is related to its angular speed:
8-4 Torque
To make an object start rotating, a force is needed; the position and direction of the force matter as well. The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.
8-4 Torque
A longer lever arm is very helpful in rotating objects.
8-4 Torque
- Torque = rotational analog of force
- r = distance from point of rotation to point where force is applied
- F = force perpendicular to r
- FA applies maximum torque Think-Pair-Share
- Problem 22: A 55-kg person riding a bike
puts all her weight on each pedal when
climbing a hill. The pedals rotate in a
circle of radius 17 cm. (a) What is the
maximum torque she exerts?
8-5 Rotational Dynamics; Torque and
Rotational Inertia
Knowing that , we see that This is for a single point mass; what about an extended object? As the angular acceleration is the same for the whole object, we can write: (8-11) (8-12)
8-6 Solving Problems in Rotational
Dynamics
- Draw a diagram.
- Decide what the system comprises.
- Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act.
- Find the axis of rotation; calculate the torques around it. 5. Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step. 6. Apply Newton’s second law for translation and other laws and principles as needed. 7. Solve. 8. Check your answer for units and correct order of magnitude.
8-6 Solving Problems in Rotational
Dynamics
