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Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ?
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Arigid uniform rod of mass m and length l is resting on a smooth horiz...
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Arigid uniform rod of mass m and length l is resting on a smooth horiz...
Given Parameters
- Mass of the rod, \( M = m \)
- Length of the rod, \( l \)
- Mass of each marble, \( m = \frac{m}{6} \)
- Speed of marbles, \( v = l \) m/s
Initial Setup
- The rod is initially at rest.
- Marbles collide inelastically at both ends of the rod.
Conservation of Momentum
- The total momentum before collision:
\[
P_{initial} = mv + mv = 2mv
\]
- After the collision, the combined system (rod + marbles) has a mass of \( M + 2m \):
\[
P_{final} = (M + 2m)V
\]
- Setting initial momentum equal to final momentum:
\[
2mv = (M + 2m)V
\]
- Substitute \( M = m \):
\[
2mv = (m + 2m)V \implies 2v = 3V \implies V = \frac{2v}{3}
\]
Moment of Inertia
- Moment of inertia of the rod about its center:
\[
I_{rod} = \frac{1}{12}Ml^2
\]
- Moment of inertia of two marbles at the ends:
\[
I_{marbles} = 2m\left(\frac{l}{2}\right)^2 = \frac{ml^2}{2}
\]
- Total moment of inertia:
\[
I_{total} = \frac{1}{12}Ml^2 + \frac{ml^2}{2} = \frac{1}{12}ml^2 + \frac{1}{2}ml^2 = \frac{7}{12}ml^2
\]
Angular Acceleration
- Torque due to the marbles:
\[
\tau = 2m \cdot \frac{l}{2} \cdot v = mlv
\]
- Using \( \alpha = \frac{\tau}{I} \):
\[
\alpha = \frac{mlv}{\frac{7}{12}ml^2} = \frac{12v}{7l}
\]
Time to Rotate Through \( \frac{\pi}{2} \)
- Using the angular kinematics equation:
\[
\theta = \omega_0 t + \frac{1}{2}\alpha t^2
\]
- Initial angular velocity \( \omega_0 = 0 \), \( \theta = \frac{\pi}{2} \):
\[
\frac{\pi}{2} = \frac{1}{2} \cdot \frac{12v}{7l} t^2
\]
- Solving for \( t \):
\[
t^2 = \frac{7\pi l}{12v} \implies t = \sqrt{\frac{7\pi l}{12v}}
\]
- Substitute \( v = l \):
\[
t = \sqrt{\
- Mass of the rod, \( M = m \)
- Length of the rod, \( l \)
- Mass of each marble, \( m = \frac{m}{6} \)
- Speed of marbles, \( v = l \) m/s
Initial Setup
- The rod is initially at rest.
- Marbles collide inelastically at both ends of the rod.
Conservation of Momentum
- The total momentum before collision:
\[
P_{initial} = mv + mv = 2mv
\]
- After the collision, the combined system (rod + marbles) has a mass of \( M + 2m \):
\[
P_{final} = (M + 2m)V
\]
- Setting initial momentum equal to final momentum:
\[
2mv = (M + 2m)V
\]
- Substitute \( M = m \):
\[
2mv = (m + 2m)V \implies 2v = 3V \implies V = \frac{2v}{3}
\]
Moment of Inertia
- Moment of inertia of the rod about its center:
\[
I_{rod} = \frac{1}{12}Ml^2
\]
- Moment of inertia of two marbles at the ends:
\[
I_{marbles} = 2m\left(\frac{l}{2}\right)^2 = \frac{ml^2}{2}
\]
- Total moment of inertia:
\[
I_{total} = \frac{1}{12}Ml^2 + \frac{ml^2}{2} = \frac{1}{12}ml^2 + \frac{1}{2}ml^2 = \frac{7}{12}ml^2
\]
Angular Acceleration
- Torque due to the marbles:
\[
\tau = 2m \cdot \frac{l}{2} \cdot v = mlv
\]
- Using \( \alpha = \frac{\tau}{I} \):
\[
\alpha = \frac{mlv}{\frac{7}{12}ml^2} = \frac{12v}{7l}
\]
Time to Rotate Through \( \frac{\pi}{2} \)
- Using the angular kinematics equation:
\[
\theta = \omega_0 t + \frac{1}{2}\alpha t^2
\]
- Initial angular velocity \( \omega_0 = 0 \), \( \theta = \frac{\pi}{2} \):
\[
\frac{\pi}{2} = \frac{1}{2} \cdot \frac{12v}{7l} t^2
\]
- Solving for \( t \):
\[
t^2 = \frac{7\pi l}{12v} \implies t = \sqrt{\frac{7\pi l}{12v}}
\]
- Substitute \( v = l \):
\[
t = \sqrt{\
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Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ?
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Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ?.
Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ?.
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Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ?, a detailed solution for Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? has been provided alongside types of Arigid uniform rod of mass m and length l is resting on a smooth horizontal table two marbles each of mass m travelling with a uniform speed v collides with two ends of the rod simultaneously and inelasticity as shown the marbles get stuck to the rod after the collision and continue to move with the rod if m is equals to m/6 and v is equals to l metre per second then the time taken by rod to rotate through π/2 is ? theory, EduRev gives you an
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