Question
Hard
Solving time: 4 mins
A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights and on the incline. The ratio is given by
(a)
(b)
(c) 1
(d)
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Text solutionVerified
Exp. (b)
When a spherical/circular body of radius rolls without slipping, its total kinetic energy is
Let be the linear velocity and be the radius for both solid sphere and solid cylinder.
Kinetic energy of the given solid sphere will be
=\frac{7}{10} m v^{2}\ldts (i)
Similarly, kinetic energy of the given solid cylinder will be
Now, from the conservation of mechanical energy,
For solid sphere,
...(iii) [using Eq. (i) ]
Similarly, for solid cylinder,
[using Eq. (ii)]
Taking the ratio of Eqs. (iii) and (iv), we get
Alternate Solution
Total kinetic energy for a rolling body without slipping can also be given as
where, is the radius of gyration.
From law of conservation,
As we know that, for solid sphere,
Similarly, for solid cylinder,
So,
When a spherical/circular body of radius rolls without slipping, its total kinetic energy is
Let be the linear velocity and be the radius for both solid sphere and solid cylinder.
Kinetic energy of the given solid sphere will be
Similarly, kinetic energy of the given solid cylinder will be
Now, from the conservation of mechanical energy,
For solid sphere,
...(iii) [using Eq. (i) ]
Similarly, for solid cylinder,
[using Eq. (ii)]
Taking the ratio of Eqs. (iii) and (iv), we get
Alternate Solution
Total kinetic energy for a rolling body without slipping can also be given as
where, is the radius of gyration.
From law of conservation,
As we know that, for solid sphere,
Similarly, for solid cylinder,
So,
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Question Text | A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights and on the incline. The ratio is given by (a) (b) (c) 1 (d) |
Updated On | Mar 24, 2022 |
Topic | System of Particles and Rotational Motion |
Subject | Physics |
Class | Class 11 |
Answer Type | Text solution:1 Video solution: 1 |
Upvotes | 164 |
Avg. Video Duration | 1 min |