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Chapter 6
Rotation of a Rigid Body about a Fixed Axis
A rigid body is defined as an object that has fixed size and shape. In other words, the relative
positions of its constituent particles remain constant. Although a perfectly rigid body does not
exist, it is a useful idealization. By โ fixed axis โ we mean that the axis must be fixed relative
to the body and fixed in direction relative to an inertia frame.
6.1 Rotational Kinematics (I)
The objects that we deal with are those which maintain a rigid shape (the mass points maintain
their relative positions) but which can change their orientation in space. They can have
translational motion, in which their center of mass moves but also rotational motion, in which we
can observe the changes in direction of a set of axes that is โglued toโ the object. Such an object
is known as a rigid body.
Figure 6.1: A point on the rotating object is located a distance r from the axis; as the object
rotates through an angle ฮธ it moves a distance s.
Because of the nice mathematical properties of expressing the measure of an angle in radians, we
will usually express angles in radians all through our study of rotations; on occasion, though, we
may have to convert to or from degrees or revolutions. Revolutions, degrees and radians are
related by:
1 revolution = 360 degrees = 2ฯ radians
6.1.1 Angular Displacement
As a rotating object moves through an angle ฮธ from the starting position, a mass point on the
object at radius r will move a distance s; s length of arc of a circle of radius r, subtended by the
angle ฮธ.
When ฮธ is in radians, these are related by
๐
๐ ๐ ๐๐ ๐๐๐๐๐๐๐ (๐๐๐)^ 6.
6.1.2 Angular Velocity
The angular position of a rotating changes with time; as with linear motion, we study the rate of
change of ฮธ with time t. If in a time period โt the object has rotated through an angular
displacement โฮธ then we define the average angular velocity (๐) for that period as
โฮธ โt 6.
A more interesting quantity is found as we let the time period โt be vanishingly small. This gives us the instantaneous angular velocity, ฯ:
๐ = lim ฮ๐กโ
โ๐ โ๐ก =
๐๐ ๐๐ก
๐๐
given time. Youโll be introduce in grade 12. Angular velocity has units of (rad/s).
In more advanced studies of rotational motion, the angular velocity of a rotating object is defined in such a way that it is a vector quantity. For an object rotating counterclockwise about a fixed axis, this vector has magnitude ฯ and points outward along the axis of rotation. For our purposes, though, we will treat ฯ as a number which can be positive or negative, depending on the direction of rotation.
6.1.3 Angular Acceleration; Constant Angular Acceleration
The rate at which the angular velocity changes is the angular acceleration of the object. If the objectโs (instantaneous) angular velocity changes by โฯ within a time period โt, then the average angular acceleration for this period is
But as you might expect, much more interesting is the instantaneous angular acceleration, defined as
๐๐
๐๐ก ๐๐ ๐กโ๐ ๐ก๐๐๐ ๐๐๐๐๐ฃ๐๐ก๐๐ฃ๐ ๐กโ๐๐ก๐๐๐ฃ๐๐ ^ ๐๐๐ ๐ก๐๐๐ก๐๐๐๐๐ข๐ ๐๐๐๐ข๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ ๐๐๐ฆ^ time.
Youโll be introduce in grade 12.
Worked Examples
6.1.1 Angular Displacement
- (a).What angle in radians is subtended by an arc that has length 1.80 m and is part of a circle of radius 1.20 m? (b) Express the same angle in degrees. (c) The angle between two radii of a circle is 0.620 rad. What arc length is subtended if the radius is 2.40 m?
6.1.2 Angular Velocity
- What is the angular speed in radians per second of (a) the Earth in its orbit about the Sun and (b) the Moon in its orbit about the Earth?
6.1.3 Angular Acceleration; Constant Angular Acceleration
- The angular position of a point on the rim of a rotating wheel is given by ฮธ = 4.0t โ 3.0t 2 + t 3 , where ฮธ is in radians if t is given in seconds. (a) What are the angular velocities at t = 2.0 s and t = 4.0 s? (b) What is the average angular acceleration for the time interval that begins at t = 2.0 s and ends at t = 4.0 s? (c) What are the instantaneous angular accelerations at the beginning and end of this time interval?
- An electric motor rotating a grinding wheel at 100 rev/min is switched off. Assuming constant negative angular acceleration of magnitude 2.00 rad/s^2 , (a) how long does it take the wheel to stop? (b) Through how many radians does it turn during the time found in (a)?
- A phonograph turntable rotating at 33 1 3 rev/ min slows down and stops in 30 s after the motor is turned off. (a) Find its (uniform) angular acceleration in units of rev/ min 2. (b) How many revolutions did it make in this time?
6. A disk, initially rotating at 120 rad/s, is slowed down with a constant angular acceleration of
magnitude 4.0 rad/s^2. (a) How much time elapses before the disk stops? (b) Through what angle
does the disk rotate in coming to rest?
7. A wheel, starting from rest, rotates with a constant angular acceleration of 2.00 rad/s^2. During a
certain 3.00 s interval, it turns through 90.0 rad. (a) how long had the wheel been turning before the start of the 3.00 s interval? (b) What was the angular velocity of the wheel at the start of the 3.00 s interval?
6.1.4 Relationship between Angular and Linear Quantities
1. What is the angular speed of a car travelling at 50 km/ hr and rounding a circular turn of radius 110 m? 2. An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10 m and, in starting, rotates according to ฮธ = 0.30t^2 , where t in seconds gives ฮธ in radians. When t = 5.0 s, what are the astronautโs (a) angular velocity, (b) linear speed, (c) tangential acceleration (magnitude only) and (d) radial acceleration (magnitude only)?
Please try to answer all the worked examples and let me see it โฆ use this telegram contact to summit your answer @akiiya. And Iโll give you all the answer before the next sub topic. Thank you
