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A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be?
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Introduction:
When a hollow sphere and a solid sphere with the same mass and radius are rolled down an inclined plane without slipping, the ratio of their speeds at the bottom can be determined. To find this ratio, we need to analyze the rotational and translational motion of both spheres.
Key Points:
1. Rotational Kinetic Energy: The rotational kinetic energy of a sphere can be calculated using the formula KE_rot = (2/5) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
2. Translational Kinetic Energy: The translational kinetic energy of a sphere can be calculated using the formula KE_trans = (1/2) * m * v^2, where m is the mass and v is the linear velocity.
3. Moment of Inertia: The moment of inertia of a hollow sphere is I_hollow = (2/3) * m * r^2, and the moment of inertia of a solid sphere is I_solid = (2/5) * m * r^2.
4. Speed Ratio: The ratio of the speeds of the hollow and solid spheres can be found by dividing their linear velocities at the bottom of the incline.
Analysis:
1. Rotational Kinetic Energy: Since both spheres are rolling without slipping, their angular velocities can be related to their linear velocities using the formula ω = v / r.
- The rotational kinetic energy of the hollow sphere is KE_rot_hollow = (2/5) * I_hollow * (v_hollow / r)^2.
- The rotational kinetic energy of the solid sphere is KE_rot_solid = (2/5) * I_solid * (v_solid / r)^2.
2. Translational Kinetic Energy: The translational kinetic energy of both spheres can be calculated using their masses and linear velocities.
- The translational kinetic energy of the hollow sphere is KE_trans_hollow = (1/2) * m * v_hollow^2.
- The translational kinetic energy of the solid sphere is KE_trans_solid = (1/2) * m * v_solid^2.
3. Total Kinetic Energy: The total kinetic energy of each sphere is the sum of its rotational and translational kinetic energies.
- The total kinetic energy of the hollow sphere is KE_total_hollow = KE_rot_hollow + KE_trans_hollow.
- The total kinetic energy of the solid sphere is KE_total_solid = KE_rot_solid + KE_trans_solid.
Conclusion:
By comparing the total kinetic energies of the hollow and solid spheres, we can determine the ratio of their speeds at the bottom of the inclined plane.
- KE_total_hollow = KE_rot_hollow + KE_trans_hollow = (2/5) * I_hollow * (v_hollow / r)^2 + (1/2) * m * v_hollow^2.
- KE_total_solid = KE_rot_solid + KE_trans_solid = (2/5) * I_solid * (v_solid / r)^2 + (1/2) * m * v_solid^2.
Simplifying these equations and dividing them, we can find the ratio of their speeds:
- Ratio of speeds = v_hollow
When a hollow sphere and a solid sphere with the same mass and radius are rolled down an inclined plane without slipping, the ratio of their speeds at the bottom can be determined. To find this ratio, we need to analyze the rotational and translational motion of both spheres.
Key Points:
1. Rotational Kinetic Energy: The rotational kinetic energy of a sphere can be calculated using the formula KE_rot = (2/5) * I * ω^2, where I is the moment of inertia and ω is the angular velocity.
2. Translational Kinetic Energy: The translational kinetic energy of a sphere can be calculated using the formula KE_trans = (1/2) * m * v^2, where m is the mass and v is the linear velocity.
3. Moment of Inertia: The moment of inertia of a hollow sphere is I_hollow = (2/3) * m * r^2, and the moment of inertia of a solid sphere is I_solid = (2/5) * m * r^2.
4. Speed Ratio: The ratio of the speeds of the hollow and solid spheres can be found by dividing their linear velocities at the bottom of the incline.
Analysis:
1. Rotational Kinetic Energy: Since both spheres are rolling without slipping, their angular velocities can be related to their linear velocities using the formula ω = v / r.
- The rotational kinetic energy of the hollow sphere is KE_rot_hollow = (2/5) * I_hollow * (v_hollow / r)^2.
- The rotational kinetic energy of the solid sphere is KE_rot_solid = (2/5) * I_solid * (v_solid / r)^2.
2. Translational Kinetic Energy: The translational kinetic energy of both spheres can be calculated using their masses and linear velocities.
- The translational kinetic energy of the hollow sphere is KE_trans_hollow = (1/2) * m * v_hollow^2.
- The translational kinetic energy of the solid sphere is KE_trans_solid = (1/2) * m * v_solid^2.
3. Total Kinetic Energy: The total kinetic energy of each sphere is the sum of its rotational and translational kinetic energies.
- The total kinetic energy of the hollow sphere is KE_total_hollow = KE_rot_hollow + KE_trans_hollow.
- The total kinetic energy of the solid sphere is KE_total_solid = KE_rot_solid + KE_trans_solid.
Conclusion:
By comparing the total kinetic energies of the hollow and solid spheres, we can determine the ratio of their speeds at the bottom of the inclined plane.
- KE_total_hollow = KE_rot_hollow + KE_trans_hollow = (2/5) * I_hollow * (v_hollow / r)^2 + (1/2) * m * v_hollow^2.
- KE_total_solid = KE_rot_solid + KE_trans_solid = (2/5) * I_solid * (v_solid / r)^2 + (1/2) * m * v_solid^2.
Simplifying these equations and dividing them, we can find the ratio of their speeds:
- Ratio of speeds = v_hollow
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A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be?
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A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be?.
A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hollow sphere and a solid sphere both having the same mass of 5kg and radius 10m are initially at rest.If they are made to roll down with same inclined plane without slipping,the ratio of their speeds when they reach the bottom of plane will be?.
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