The kinetic energy of a particle of mass m1 moving with a speed V is same as the kinetic energy of a solid sphere of mass m2 rolling on the plane surface. If the speed of the Centre of the sphere is also V,then m1/m2 is:
\(\frac{7}{10}\)
\(\frac{1}{2}\)
\(\frac{5}{7}\)
\(\frac{7}{5}\)
\(\frac{2}{3}\)
The Correct Option is D
Approach Solution - 1
Given:
- Particle KE: \( \frac{1}{2} m_1 V^2 \).
- Rolling sphere KE: Translational + Rotational = \( \frac{1}{2} m_2 V^2 + \frac{1}{2} I \omega^2 \).
- For a solid sphere: \( I = \frac{2}{5} m_2 R^2 \) and \( \omega = \frac{V}{R} \).
Step 1: Equate Kinetic Energies
\[ \frac{1}{2} m_1 V^2 = \frac{1}{2} m_2 V^2 + \frac{1}{2} \left(\frac{2}{5} m_2 R^2\right) \left(\frac{V}{R}\right)^2 \]
Simplify:
\[ \frac{1}{2} m_1 V^2 = \frac{1}{2} m_2 V^2 + \frac{1}{5} m_2 V^2 \]
\[ m_1 V^2 = m_2 V^2 \left(1 + \frac{2}{5}\right) = \frac{7}{5} m_2 V^2 \]
\[ \frac{m_1}{m_2} = \frac{7}{5} \]
Conclusion:
The mass ratio is \(\frac{7}{5}\).
Approach Solution -2
1. Write the kinetic energy of the particle:
The kinetic energy (KE) of a particle of mass \(m_1\) moving with speed \(V\) is given by:
\[KE_1 = \frac{1}{2}m_1V^2\]
2. Write the kinetic energy of the rolling sphere:
The kinetic energy of a rolling solid sphere of mass \(m_2\) and radius \(r\) with a center-of-mass speed \(V\) is the sum of its translational and rotational kinetic energies:
\[KE_2 = \frac{1}{2}m_2V^2 + \frac{1}{2}I\omega^2\]
where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
3. Substitute the moment of inertia and angular velocity:
For a solid sphere, \(I = \frac{2}{5}m_2r^2\). Since the sphere rolls without slipping, the angular velocity is related to the linear velocity by \(\omega = \frac{V}{r}\). Substituting these values:
\[KE_2 = \frac{1}{2}m_2V^2 + \frac{1}{2}(\frac{2}{5}m_2r^2)(\frac{V}{r})^2\]
\[KE_2 = \frac{1}{2}m_2V^2 + \frac{1}{5}m_2V^2 = \frac{7}{10}m_2V^2\]
4. Set the kinetic energies equal and solve for the mass ratio:
Given that \(KE_1 = KE_2\):
\[\frac{1}{2}m_1V^2 = \frac{7}{10}m_2V^2\]
Canceling out the common terms \(V^2\):
\[\frac{1}{2}m_1 = \frac{7}{10}m_2\]
\[\frac{m_1}{m_2} = \frac{7}{10} \times 2 = \frac{7}{5}\]
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Concepts Used:
Kinetic energy
Kinetic energy of an object is the measure of the work it does as a result of its motion. Kinetic energy is the type of energy that an object or particle has as a result of its movement. When an object is subjected to a net force, it accelerates and gains kinetic energy as a result. Kinetic energy is a property of a moving object or particle defined by both its mass and its velocity. Any combination of motions is possible, including translation (moving along a route from one spot to another), rotation around an axis, vibration, and any combination of motions.
