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EQUATIONS OF MOTIONS ALONG A CIRCULAR PATH

Consider a body, which is moving along a circular path.

Let ω0 = Final angular velocity of the body in radians per second

ω = Final angular velocity in radians per second

t = Time in second during which angular velocity changes from ω0 to ω

Θ = Angular displacement or angle traversed in radians

α= Angular acceleration in radians/s2

From Art. 5.6.3, we know that angular acceleration (α) is the rate of change of angular velocity. Hence

Problem 5.28. A body is rotating with an angular velocity of 5 radians / s. After 4 seconds, the angular velocity of the body becomes 13 radians / s. Determine the angular acceleration of the body.

Sol. Given:

Initial angular velocity, ω0 = 5 rad/s

Final angular velocity, ω = 13 rad/s

Problem 5.32. A flywheel is rotating at 200 r.p.m. and after 10 seconds it is rotating at 160 r.p.m. If the retardation is uniform, determine member of revolution made by the flywheel and the time taken. by the flywheel before it comes to rest from the speed of 200 r.p.m.

Sol. Given:

Initial speed,  N0 = 200 r.p.m.

(i) Number of revolutions made by flywheel before it stops.

First determine the uniform retardation. Using equation (5.15), we have

To find the number of revolutions made by the flywheel before coming to rest from 200 r.p.m. (or 20.94 rad/s), we have

Initial angular velocity, ω0 = 20.94 rad/s

Final angular velocity, ω = 0

Angular retardation,   = – .419 rad/s2

Now using the equation (5.17),

 (ii) Time taken by wheel before it comes to rest from 200 r.p.m.

Initial angular velocity, ω0 == 20.94 rad/s

Final angular velocity, ω = 0

Angular retardation, α = – 0.419 rad/s2

Using the equation (5.15), ω = ω0 + at

0= 20.94 + (-.0419) × t

Problem 5.33. The angle of rotation of a body is given by the equation

ω = 2t3 - 5t2 + 8t + 6

where ω is expressed in radians and t in seconds. Determine :

(i)                           angular velocity and

(ii)                        angular acceleration oitlu: body

when t = 0 and t = 4 seconds.

Sol. Given:

Angular displacement θ = 2t3 – 5t2 + 8t + 6.

Angular velocity (ω) is obtained differentiating the above equation with respect to time (t)

When t = 0

Angular velocity (ω) and angular acceleration (α), when t = 0, are obtained by substituting t = 0 in equations (i) and (ii) respectively,

ω = 6 × 0 – 10 × 0 +8 = 0 – 0 + 8 = 8 radian/s Ans.

α = 12 × 0 – 10 = -10 radians/s2 Ans.

When t = 4 seconds

Substituting t = 4 seconds in equations (i) and (ii), we get

ω = 6 × 42 – 10 × 4 + 8 = 96 – 40 + 8 = 64 radian/s Ans.

α = 12 × 4 – 10 = 48 – 10 = 38 radian/s2 Ans.

Problem 5.34. The initial angular velocity of a rotating body is 2 rad/s and initial
angular acceleration is zero. The rotation of the body is according
to the relation ω = 3t2-3, Find:

(i)                           angular velocity and

(ii)                        angular displacement when t = 5 seconds. Consider the angular displacement in radians and time in seconds.

Sol. Given:

Initial angular velocity, ω0 =2 rad/s

This means that when t = 0, the angular velocity is 2 rad/s

Initial angular acceleration, ω = 0

The law of rotation, ω = 3t2 – 3.

Find: (i) angular velocity (ω) and (ii) angular displacement e when t = 5 s.

(i) Angular velocity (ω)

where C is a constant of integration. Its value is obtained from the given condition i.e., ω = 2 rad/s when t = 0.

Substituting these values in equation (i), We get

2 = 0 – 0 + C             or             C = 2

Substituting the value of C in equation (z),

ω = t3- 3t + 2

When t = 5 s, angular velocity becomes

ω = 53 – 3 × 5 + 2 = 125 – 15 + 2 = 112 rad/s.  Ans.

(ii) Angular displacement (θ)

Problem 5.35.The angle of rotation of a body is given as a function of time by the equation,

θ = θ0 + at + bt2

where θ0 initial angular displacement, a and b are constants.

Obtain general expressions for:

(i)                           the angular velocity and

(ii)                        the angular acceleration of the the body

If the initial angular velocity be 3Πradian per second and after two seconds the angular velocity is 8Π radian per second, determine the constants a and b.

Sol. Given:

θ = θ0 + at + bt2.

Angular velocity (ω) is obtained by differentiating the above equation with respect to time (t).

(i) When t = 0, ω= 3Π

Substituting these values in equation (i), we get

3Π = a + 2b × 0 = a

a = 3Π. Ans.

(ii) When t = 2 seconds, ω = 8Π radians.

Substituting these values in ω = a + 2bt, we get

3Π = a + 2b × 2 = a + 4b = 3 Π + 4b

4b = 8Π – 3Π = 5Π

(b) General expression for angular acceleration

Angular acceleration (α) is obtained by differentiating the equation (i) with respect to time (t).