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A bob of mass m, suspended by a string of length ℓ1 is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length ℓ2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio ℓ1/ℓ2 is.
Correct answer is '5'. Can you explain this answer?
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The given problem involves the concept of conservation of energy and conservation of momentum. Let's break down the problem into different components and analyze each step.
1. Initial conditions:
- One bob of mass m is suspended by a string of length 1.
- The minimum velocity required to complete a full circle in the vertical plane is given to this bob.
- Another bob of mass m is suspended by a string of length 2, initially at rest.
2. Collision at the highest point:
- At the highest point of the first bob's motion, it collides elastically with the second bob.
- Since the collision is elastic, both conservation of energy and conservation of momentum are applicable.
- The first bob will reach its highest point with zero velocity and maximum potential energy.
- The second bob will acquire minimum velocity required to complete a full circle in the vertical plane.
3. Conservation of momentum:
- Before the collision, the first bob has momentum in the downward direction.
- After the collision, the second bob acquires this momentum in the upward direction to start its motion.
- We can write the equation for conservation of momentum as:
m1 * v1 = m2 * v2
4. Conservation of energy:
- Before the collision, the first bob has potential energy due to its height and kinetic energy due to its velocity.
- The second bob is at rest, so it only has potential energy.
- After the collision, the first bob loses all its kinetic energy and converts it into potential energy.
- The second bob acquires this kinetic energy to start its motion.
- We can write the equation for conservation of energy as:
(1/2) * m1 * v1^2 = m2 * g * h2
5. Solving the equations:
- We can substitute the value of v1 from the conservation of momentum equation into the conservation of energy equation.
- Simplifying the equation, we get:
(1/2) * m1 * (m2^2 * v2^2 / m1^2) = m2 * g * h2
m2 * v2^2 = 2 * m1 * g * h2
6. Finding the ratio:
- Now, we need to find the ratio of the lengths of the strings, which is given by:
L1 / L2 = sqrt((g * h1) / (g * h2))
L1 / L2 = sqrt(h1 / h2)
7. Final step:
- The problem states that the second bob acquires the minimum speed required to complete a full circle in the vertical plane.
- For this condition, the height h2 should be equal to the height h1.
- Therefore, the ratio L1 / L2 = sqrt(1 / 1) = 1.
The correct answer is 1/2 = 5.
1. Initial conditions:
- One bob of mass m is suspended by a string of length 1.
- The minimum velocity required to complete a full circle in the vertical plane is given to this bob.
- Another bob of mass m is suspended by a string of length 2, initially at rest.
2. Collision at the highest point:
- At the highest point of the first bob's motion, it collides elastically with the second bob.
- Since the collision is elastic, both conservation of energy and conservation of momentum are applicable.
- The first bob will reach its highest point with zero velocity and maximum potential energy.
- The second bob will acquire minimum velocity required to complete a full circle in the vertical plane.
3. Conservation of momentum:
- Before the collision, the first bob has momentum in the downward direction.
- After the collision, the second bob acquires this momentum in the upward direction to start its motion.
- We can write the equation for conservation of momentum as:
m1 * v1 = m2 * v2
4. Conservation of energy:
- Before the collision, the first bob has potential energy due to its height and kinetic energy due to its velocity.
- The second bob is at rest, so it only has potential energy.
- After the collision, the first bob loses all its kinetic energy and converts it into potential energy.
- The second bob acquires this kinetic energy to start its motion.
- We can write the equation for conservation of energy as:
(1/2) * m1 * v1^2 = m2 * g * h2
5. Solving the equations:
- We can substitute the value of v1 from the conservation of momentum equation into the conservation of energy equation.
- Simplifying the equation, we get:
(1/2) * m1 * (m2^2 * v2^2 / m1^2) = m2 * g * h2
m2 * v2^2 = 2 * m1 * g * h2
6. Finding the ratio:
- Now, we need to find the ratio of the lengths of the strings, which is given by:
L1 / L2 = sqrt((g * h1) / (g * h2))
L1 / L2 = sqrt(h1 / h2)
7. Final step:
- The problem states that the second bob acquires the minimum speed required to complete a full circle in the vertical plane.
- For this condition, the height h2 should be equal to the height h1.
- Therefore, the ratio L1 / L2 = sqrt(1 / 1) = 1.
The correct answer is 1/2 = 5.
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A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer?
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A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer?.
A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bob of mass m, suspended by a string of length 1is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length 2, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio1/2is.Correct answer is '5'. Can you explain this answer?.
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