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A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is K. If radius of the ball be R, then the fraction of total energy associated with its rotational energy will be
- a)K2+R2/R2
- b)K2/R2
- c)K2/K2+R2
- d)R2/K2+R2
Correct answer is option 'C'. Can you explain this answer?
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A ball rolls without slipping. The radius of gyration of the ball abou...
Solution:
Given, radius of gyration of the ball about an axis passing through its centre of mass = K
Radius of the ball = R
Total energy associated with the ball = E
Fraction of total energy associated with its rotational energy = ?
Let the ball be rolling with a linear velocity v and angular velocity ω.
The kinetic energy of the ball can be written as:
E = 1/2 mv² + 1/2 Iω²
where m is the mass of the ball and I is the moment of inertia of the ball about an axis passing through its centre of mass.
Since the ball is rolling without slipping, we have:
v = Rω
Substituting this in the equation for kinetic energy, we get:
E = 1/2 mv² + 1/2 I(v/R)²
= 1/2 mv² + 1/2(I/R²)v²
= 1/2 mv² + 1/2(mK²)v²
Therefore, the fraction of total energy associated with the rotational energy is given by:
(1/2(mK²)v²) / (1/2 mv² + 1/2(mK²)v²)
= K² / (R² + K²)
= K² / (K² + R²)
Therefore, the correct option is (C) K²/(K²+R²).
Given, radius of gyration of the ball about an axis passing through its centre of mass = K
Radius of the ball = R
Total energy associated with the ball = E
Fraction of total energy associated with its rotational energy = ?
Let the ball be rolling with a linear velocity v and angular velocity ω.
The kinetic energy of the ball can be written as:
E = 1/2 mv² + 1/2 Iω²
where m is the mass of the ball and I is the moment of inertia of the ball about an axis passing through its centre of mass.
Since the ball is rolling without slipping, we have:
v = Rω
Substituting this in the equation for kinetic energy, we get:
E = 1/2 mv² + 1/2 I(v/R)²
= 1/2 mv² + 1/2(I/R²)v²
= 1/2 mv² + 1/2(mK²)v²
Therefore, the fraction of total energy associated with the rotational energy is given by:
(1/2(mK²)v²) / (1/2 mv² + 1/2(mK²)v²)
= K² / (R² + K²)
= K² / (K² + R²)
Therefore, the correct option is (C) K²/(K²+R²).
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A ball rolls without slipping. The radius of gyration of the ball abou...
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