Chapter 31 : Kinetics of Motion of Rotation 649
Example 31.17. A solid uniformly thick wheel of radius 1 m and mass 40 kg is released
with no initial velocity at the top of an inclined plane, which makes an angle of 30° with the hori-
zontal. It rolls down without slipping. Determine (i) the minimum value of coefficient of friction, (ii)
the velocity of the centre of the wheel after it has travelled a distance 4 m down the inclined plane.
Solution. Given: Radius of the wheel (r) = 1 m; *Mass of the wheel (m) = 40 kg and inclina-
tion of plane (θ) = 30°
(i) Minimum value of coefficient of friction
We know that for a uniformly thick wheel,
k^2 = 0.5 r^2and minimum value of coefficient of friction,
μ= 22 2222tan tan 30 0.5774
0.192
0.5^3
0.5kr rr
krθ°
===
++Ans.(ii) Velocity of the centre of the wheel after it has travelled a distance of 4 m
Let v= Velocity of the centre of the wheel.
We know that acceleration of the wheel when it rolls down the plane,a= 22 22 222sin 9.8 sin 30 9.8 0.5
3.27 m/s
0.5 1.5g
kr rr
rrθ°×
===
++and v^2 =u^2 + 2as = 0 + 2 × 3.27 × 4 = 26.16
v= 5.1 m/s Ans.EXERCISE 31.3
- A solid right circular roller of mass 15 kg is being pulled by another body of mass 15 kg
along a horizontal plane as shown in Fig. 31.18.
Fig. 31.18.
Find the acceleration of the roller, assuming that there is no slipping of the roller and
string is inextensible. [Ans. 2.8 m/s^2 ]- A solid sphere of mass (m) and radius (r) rolls down a plane inclined at an angle (θ) with
the horizontal. Find the acceleration of the sphere. [Ans. 7 sin θ] - A solid cylinder is placed on a plane inclined at 13° 18 with the horizontal and allowed
to roll down without slipping and with its axis horizontal. Find the acceleration of the
cylinder. [Ans. 1.5 m/s^2 ] - Find the time taken by a solid cylinder of radius 600 mm and initially at rest to roll down
a distance 19.2 m on a plane inclined at 30° with the horizontal. Take g = 9.81 m/s^2.
[Ans. 3.5 s]
- Superfluous data.