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A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)?
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Understanding the Problem
A small block of mass "m" is projected horizontally from a smooth hemisphere of radius R with speed "v". The goal is to determine the minimum speed "v" required for the block to not slide off the hemisphere, meaning it must leave the surface at the top.
Forces Acting on the Block
- When the block is at the top of the hemisphere, the gravitational force acts downwards.
- The normal force acts perpendicular to the surface of the hemisphere.
Condition for Leaving the Surface
- To ensure that the block does not slide down, the normal force must be zero when the block is at the top.
- At this point, the only force acting on the block is its weight, which provides the necessary centripetal force to maintain circular motion.
Applying Energy Conservation
- The total mechanical energy (kinetic + potential) is conserved.
- Initially, the block has kinetic energy due to its horizontal speed, and potential energy due to its height.
Calculating Minimum Speed
- At the top of the hemisphere, the potential energy is maximum, and the block must have enough kinetic energy to maintain circular motion.
- The relationship between gravitational potential energy and kinetic energy leads us to derive the minimum speed required.
Final Expression
- The minimum speed "v" at the top for the block to not slide off can be determined using the equations of motion and energy conservation.
- The final expression for the minimum speed "v" is v = √(gR), where g is the acceleration due to gravity.
Conclusion
By ensuring that the normal force is zero at the top and applying conservation principles, we find the minimum speed necessary for the block to remain on the hemisphere. This analysis highlights the balance between gravitational forces and centripetal requirements for motion on curved surfaces.
A small block of mass "m" is projected horizontally from a smooth hemisphere of radius R with speed "v". The goal is to determine the minimum speed "v" required for the block to not slide off the hemisphere, meaning it must leave the surface at the top.
Forces Acting on the Block
- When the block is at the top of the hemisphere, the gravitational force acts downwards.
- The normal force acts perpendicular to the surface of the hemisphere.
Condition for Leaving the Surface
- To ensure that the block does not slide down, the normal force must be zero when the block is at the top.
- At this point, the only force acting on the block is its weight, which provides the necessary centripetal force to maintain circular motion.
Applying Energy Conservation
- The total mechanical energy (kinetic + potential) is conserved.
- Initially, the block has kinetic energy due to its horizontal speed, and potential energy due to its height.
Calculating Minimum Speed
- At the top of the hemisphere, the potential energy is maximum, and the block must have enough kinetic energy to maintain circular motion.
- The relationship between gravitational potential energy and kinetic energy leads us to derive the minimum speed required.
Final Expression
- The minimum speed "v" at the top for the block to not slide off can be determined using the equations of motion and energy conservation.
- The final expression for the minimum speed "v" is v = √(gR), where g is the acceleration due to gravity.
Conclusion
By ensuring that the normal force is zero at the top and applying conservation principles, we find the minimum speed necessary for the block to remain on the hemisphere. This analysis highlights the balance between gravitational forces and centripetal requirements for motion on curved surfaces.
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A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)?
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A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)? 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 A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)?.
A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)? 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 A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A small block of mass "m" is projected horizontally from the top of a smooth hemisphere of radius R with speed "v". Find the minimum value of "v" so that the block does not slide on the hemisphere (i.e. it leaves the surface at the top itself)?.
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