The Rigid Body. A body in which the distance between any two points does not change due to the application of external forces is called a rigid body (figure 1).
Motion of a Rigid Body. Various types of motions of a rigid body can be grouped mainly into three categories viz.
Translation, fixed axis rotation and the general plane motion.
A motion is said to be a translation if any straight line inside the body keeps the same direction during the motion. It may also be observed that in a translation all the particles forming the body move along parallel paths. If these paths are straight lines, the motion is said to be rectilinear translation (figure 2); if the paths are curved lines, the motion is a curvilinear translation (figure 3).
Position of a point in the rigid body. The position of any point B of the rigid body in translation is described with respect to another point A of the rigid body (figur4). The position of B with respect to A is denoted by the position vector vB/A. Using vector addition.
rB = rA + rB/A ...(1)
Velocity of a point in the rigid body. Relationship between the instantaneous velocites of A and B is given by differentiating eq. (1) with respect to time.
vB = v A ......(2) (Since0)
Therefore, all the points of a rigid body have same velocity.
Acceleration. Taking time derivative of equation (2) yields a relationship between accelerations of A and B.
aA= aB . . . . . . . . . . . . . (3)
Therefore, all the points of a rigid body have same acceleration.
3. Rotation About A Fixed Axis
In this type of motion, the particles forming the rigid body move in parallel planes along circles centered on the same fixed axis (figure 5).
Rotation should not be confused with of curvilinear translation. For example, the plate shown in figure 6(a) is in curvilinear translation, with all its particles moving along parallel circles, while the plate shown in figure 6(b) is in rotation, with all its particles moving along concentric circles having center of the point of suspension.
Angular Position. At the instant shown, the angular position of the radial line r is defined by the angle measured between a fixed reference line and r. Here r extends normally from the axis of rotation at point O to a point P in the body (figure 7).
Angular Displacement. The change in angular position d is called the angular displacement. d is a vector which is measured in degrees, radians or revolutions (1 rev = 2 rad). Direction of d is given by right hand rule (figure 7.)
Angular Velocity. The time derivative of d is called the angular velocity
is an axial and vector has a direction always along the axis of rotation. i.e. in the same direction asd (figure7). It is measured in rad/sec.
Angular Acceleration. The angular acceleration a measures the time rate of change of the angular velocity.
or = ...(6) (since = )
Eliminating dt from equation (4) and equation (5) yields
d = d ...(7)
Constant Angular Acceleration. If the angular acceleration of the body is constant i.e. = c and and are collinear the integration of equations (4), (5) and (7) gives
= 0+ ct ...(8)
= 0 + 0t + ct2 ...9)
2 = + 22 ( – 0 ) ...(10)
here 0 and 0 are the initial values of the body’s angular position and angular velocity respectively.
4. Motion of Any point P of the Rigid Body in fixed axis Rotation
As the rigid body rotates, the point P travels along circular path of radius r and centred at point O (figure 8), which lies on the axis of rotation.
Velocity of Point P. In scaler form the velocity of point P of rigid body is given as
v = r ...(11)
the direction of v is tangent to the circular path (figure .8). In vector notations, the velocity of point P is given by
v = × r ...(12)
Acceleration of Point P. The acceleration of point P is expressed in normal and tangential components (figure 9)
at = r ... (13)
an = 2 r ... (14)
In vector notations the total acceleration a of the point P is expressed as
a = at + an ..(15)
or a = × r – 2 r ...16)
Sample Problem1. (Rotation about fixed axis)
A cord is wrapped around a wheel which is initially at rest (figure A). If a force is applied to the cord and gives it an acceleration a = (4t)m/s2, where t is in seconds, determine as a function of time (a) the angular velocity of the wheel, and (b) the angular position of line OP in radians.
Solution: Part (a). The wheel is subjected to rotation about a fixed axis passing through point O. Thus, point P on wheel has motion about a circular path, and therefore the acceleration of this point has both tangential and normal components. In particular, the tangential component is (ap)t = (4t)m/s2, since the cord is connected to the wheel and tangent to it at P. Hence the angular acceleration of the wheel is
Using this result, the wheel’s angular velocity w can now be determined from = d/dt,* since this equation relates, t and . Integrating, with the initial condition that = 0 at t = 0, yields
Part (b). Using this result, the angular position q of the radial line OP can be computed from = d/dt, since this equation relates ,and t. Integrating, with the initial condition = 0 at t = 0, we have
Sample Problem 2. (Rotation about fixed axis)
Disk A (figure A) starts from rest and through the use of motor begins to rotate with a constant angular acceleration of A = 2 rad/s2. If no slipping occurs between the disks, determine the angular velocity and angular acceleration of disk B just after A turns 10 revolutions.
Solution: First we will convert the 10 revolutions to radians. Since there are 2 rad to one revolution, then
Since A is constant, the angular velocity of A is then
As shown in figure B the speed of the contacting point P on the rim of A is
(+ ) vP = A rA = (15.9 rad/s)(0.6 m) = 9.54 m/s
The velocity is always tangent to the path of motion; and since no slipping occurs between the disks, the speed of point P' on B is the same as the speed of P on A*. The angular velocity of B is therefore
The tangential components of acceleration of both disks are also
equal, since the disks are in contact with one another. Hence, from figure C.
(ap)t = (ap')t
Its is important notice that the normal components of acceleration (ap)nand (ap')n act in opposite directions, since the paths of motion for both points are different. Furthermore, (ap)n (ap')n since the magnitudes of these components depend on both the radius and angular velocity of each disk, i.e., (ap)n = rAand
(ap)n = rB. Consequently, ap ap,.
A flywheel 0.4 m in diameter is brought uniformly from rest up to a speed of 240 rpm in 2
sec. What is the velocity of a point on the rim 1 s after starting from rest ?
(a) 0.2 m/s
(b) 0.4 m/s
(c) 0.6 m/s
(d) 0.8 m/s
A ball rolls 2 m across a flat car in a direction perpendicular to the path of the car. In the same time interval during which the ball is rolling, the car moves at a constant speed on the horizontal straight track for a distance of 2.5 m. What is the absolute displacement of the ball ?
(a) 3.2 m
(b) 1.6 m
(c) 0.8 m
(d) 0.4 m
A rigid body is rotating at 180 rev/min about a line i – 2j – 2k. The origin is on the line. What is the magnitude of the linear velocity of a point (1 m, 1 m, 1 m) ?
Load B is connected to a double pulley by one of the two inextensible cables (figure A). The motion of the pulley is controlled by the cable C, which has a constant acceleration of 0.225 m/s
and an initial velocity of 0.3 m/s both directed to the right. What is the number of revolutions executed by the pulley in 2 s ?
A rigid body is rotating at 5 rad/s about an axis through origin and with direction cosines 0.4, 0.6 and 0.8 with respect to x, y and z-axis respectively. What is the magnitude of velocity of a point in the body defined by the position vector r = – 2i + 3j – 4k with respect to the origin.
(d) None of these
For the system of connected bodies (figure B) the initial angular velocity of the compound pulley B is 6 rad per sec counterclockwise and weight D is decelerating at the constant rate of 4 cm/s2
. What distance will weight A travel before coming to rest ?
(a) 6.5 cm
(b) 9.5 cm
(c) 11.5 cm
(d) 13.5 cm
When the angular velocity of a 4 cm diameter pulley is 3 rad per sec, the total acceleration of a point on its rim is 30 cm/s2
. Determine the angular acceleration of the pulley at this instant.
(a) 12 rad/s2
(b) 10 rad/s2
(b) 8 rad/s2
(d) 6 rad/s2
Determine the horizontal component of the acceleration of point B on the rim of the flywheel (figure C). At the given position,
= 4 rad per sec and= 12 rad per sec2
, both clockwise.
A pulley has a constant angular acceleration of 12 rad per sec2
. When the angular velocity is 3 rad per sec, the total acceleration of a point on the rim of the pulley is 10 m/s2
. Compute the diameter of the pulley ?
(a) 1/3 m
(b) 2/3 m
(c) 1 m
(d) 4/3 m
The step pulleys are connected by a crossed belt. If the angular acceleration of C is 2 rad per sec2, what time is required for A to travel 64 ft from rest ? D move while A moves 100 ft.
(a) 2 sec
(b) 3 sec
(c) 4 sec
(d) 6 sec