Rotational Motion
By a rigid body we mean a body with a definite shape that doesn’t change, so that the particles composing it stay in fixed positions relative to one another.
By purely rotational motion, we mean that all points in the body move in circles, such as point P.
Angular Quantities
Radians One radian is defined as an angle subtended by an arc whose length is equal to the radius. In general, any angle may be defined in radians as the length of arc divided by radius
Angular (Rotational) Velocity
Direction of Rotational Velocity
Angular (Rotational) Acceleration
Direction of Rotational Acceleration
Kinematic Equations for Uniformly Accelerated Rotational Motion
Question On a rotating platform, if you sit halfway between the rotating axis and the outer edge and have a rotational speed of 20 RPM and tangential speed of 2 m/s, what will be the rotational and tangential speeds of your friend, who sits on the outer edge?
Railroad Train Wheels
Rotational Inertia
Just as object at rest tends to stay at rest and an object in motion tends to remain moving in a straight line, an object rotating about an axis tends to remain rotating about the same axis unless interfered with some external influence.
The property of an object to resist changes in its rotational state of motion is called rotational inertia.
Unlike the linear motion, rotational inertia depends on the distribution of the mass relative to the axis of rotation.
The greater the distance between the bilk of an object’s mass and its axis of rotation, the greater the rotational inertia.
Tightrope Walker and Other Examples
Because of rotational inertia, a solid cylinder starting from rest will roll down an incline faster than a ring or a hoop.
Questions Consider balancing a hammer upright on the tip of your finger. If the head of hammer is heavy and the handle long, would it be easier to balance with the end of the handle on your finger so that the head is at the top, or the other way around with the head at your fingertip and the end of the handle on the top? Consider a pair of meter sticks standing nearly upright against the wall. If you release them, they’ll rotate to the floor in the same time. But what if one has a massive hunk of clay stuck to its top end? Will it rotate to the floor in a longer or shorter time?
A figure skater is spinning with her arms held straight out A figure skater is spinning with her arms held straight out. Which has greater rotational speed, her shoulders or her fingertips? Why? You are looking down on a merry-go-round and observe the it is rotating clockwise. What is the direction of the merry-go-round rotational velocity? If the merry-go-round is slowing down, what is the direction of its rotational acceleration? What is the direction of rotational velocity of Earth?
Earth rotational speed is slowing due to tidal influences of the Sun and Moon. What is the direction of Earth’s rotational acceleration? Two flywheels have the same mass but one of them is much thinner than the other so that its radius is twice that of the other one. If both flywheels are spinning about their axes at the same rate, which one would be harder to stop? Why?