Prepared by:
Engr. Brenz Eduard C. Ilagan
Faculty, College of Engineering
University of Batangas Main Campus
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
Session 4
Rotation
In this session, you will get to know more about the concepts of dynamics of rigid bodies and
other engineering science. By the end of this session you should be able to:
1. Analyze motion of a rigid body in a circular path with its center on a fixed straight line
called the axis of rotation.
2. Understand the kinematics characteristics of rotation which are the following: angular
displacement, angular velocity and angular acceleration.
Lecture:
ROTATION ABOUT A FIXED AXIS
When a body rotates about a fixed axis, any point P located in the body travels along a circular path. To
study this motion it is first necessary to discuss the angular motion of the body about the axis.
Angular Motion
Since a point is without dimension, it cannot have angular motion.
Only lines or bodies undergo angular motion. For example,
consider the body shown in Fig. and the angular motion of a radial
line r located within the shaded plane.
Angular Position
At the instant shown, the angular position of r is defined by the angle Ө,
measured from a fixed reference line to r.
Angular Displacement
The change in the angular position, which can be measured as a
differential dӨ, is called the angular displacement.
This vector has a magnitude of d Ө, measured in degrees, radians, or
revolutions, where 1 rev = 2 π rad. Since motion is about a fixed axis, the
direction of dӨ is always along this axis. Specifically, the direction is
determined by the right-hand rule; that is, the fingers of the right hand are
curled with the sense of rotation, so that in this case the thumb, or dӨ,
points upward, Fig. 16–4a. In two dimensions, as shown by the top view of
the shaded plane, Fig. 16–4b, both Ө and dӨ are counterclockwise, and so
the thumb points outward from the page.
Angular Velocity
The time rate of change in the angular position
is called the angular velocity ω(omega). Since dӨ occurs during an
instant of time dt, then,
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
This vector has a magnitude which is often measured in rad/s. It is
expressed here in scalar form since its direction is also along the axis of
rotation, Fig. 16–4a. When indicating the angular motion in the shaded
plane, Fig. 16–4b, we can refer to the sense of rotation as clockwise or
counterclockwise. Here we have arbitrarily chosen counterclockwise
rotations as positive and indicated this by the curl shown in parentheses
next to Eq. 16–1. Realize, however, that the directional sense of ω is
actually outward from the page.
Angular Acceleration
The angular acceleration α(alpha) measures the time rate of change of the angular velocity. The magnitude
of this vector is
Using Eq. 16–1, it is also possible to express α as
The line of action of α is the same as that for ω, Fig. 16–4a; however, its sense of direction depends on
whether ω is increasing or decreasing. If ω is decreasing, then α is called an angular deceleration and
therefore has a sense of direction which is opposite to ω. By eliminating dt from Eqs. 16–1 and 16–2, we
obtain a differential relation between the angular acceleration, angular velocity, and angular
displacement, namely,
CONSTANT ANGULAR ACCELERATION
If the angular acceleration of the body is constant, then the following equation, when integrated, yield a set
of formulas which relate the body’s angular velocity, angular position, and time. These equations are
similar to one used for rectilinear motion. The results are:
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
Motion of Point P. As the rigid body in Fig. 16–4c rotates, point P
travels along a circular path of radius r with center at point O. This path
is contained within the shaded plane shown in top view, Fig. 16–4d.
Position and Displacement. The position of P is defined by the
position vector r, which extends from O to P. If the body rotates du then P
will displace ds = r dӨ.
Velocity. The velocity of P has a magnitude which can be found by
dividing ds = r du by dt so that
As shown in Figs. 16–4c and 16–4d, the direction of v is tangent to the
circular path.
Acceleration
The acceleration of P can be expressed in terms of its normal and tangential components. Applying
Equation,
The tangential component of acceleration represents the time rate of change
in the velocity’s magnitude. If the speed of P is increasing, then at acts in the same
direction as v; if the speed is decreasing, at acts in the opposite direction of v;
and finally, if the speed is constant, at is zero.
The normal component of acceleration represents the time rate of
change in the velocity’s direction. The direction of an is always toward O,
the center of the circular path.
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
ILLUSTRATIVE PROBLEM NO. 1
The rim of a 50in wheel on a brake shoe testing machine has a speed of 60mph when the brake is
dropped. It comes to rest after the rim has traveled a linear distance of 600ft. What are the constant
angular acceleration and the number of revolutions the wheel makes in coming to rest?
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
ILLUSTRATIVE PROBLEM NO. 2
When the angular velocity of a 4ft diameter pulley is 3 rad/sec., the total acceleration of a point on its rim
is 30fps^2. A.) Determine the normal acceleration of the pulley at this instant. B.) Determine the tangential
acceleration of the pulley at this instant. C.) Determine the angular acceleration of the pullet at this
instant.
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
ILLUSTRATIVE PROBLEM NO. 3
The rotation of a pulley is defined by the relation is defined by the rotation Ө = 2t^4 – 30t^2+6 where Ө is
measured in radians and t is in seconds. a.) Compute the value of the angular displacement Ө when t =
4sec. b.) Compute the value of the angular velocity at the instant t = 4sec. c.) Compute the value of the
angular acceleration at the instant t = 4 sec.
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN
ILLUSTRATIVE PROBLEM NO. 4
The angular acceleration of a pulley which will rotate from rest is increased uniformly from zero to 12 rad/s^2
during 4 sec., and then uniformly decreased to 4 rad/s^2 during the next 3 sec. a.) Compute the angular
velocity at the end of 7 sec. b.) Compute the angular displacement at the end of 4 sec.
DYNAMICS OF RIGID BODIES | ENGR. BRENZ EDUARD C, ILAGAN