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A simple pendulum with length l and bob of mass m is executing S.H.M of small amplitude a. The expression for maximum tension in the string will be
- a)mg
- b)
- c)
- d)mg[1 + (a/l) ]
Correct answer is option 'B'. Can you explain this answer?
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A simple pendulum with length l and bob of mass m is executing S.H.M o...
The tension in the string would be maximum when bob will be passing mean position.
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A simple pendulum with length l and bob of mass m is executing S.H.M o...
The maximum tension in the string of a simple pendulum executing simple harmonic motion can be determined by considering the forces acting on the bob at the extreme points of its motion.
1. Forces acting on the bob:
At the extreme points of its motion, the bob experiences two forces:
- The weight force (mg) acting vertically downward.
- The tension force (T) acting along the length of the string.
2. Maximum tension:
To find the maximum tension, we need to consider the point where the tension is maximum, which is at the extreme points of the motion, when the bob is farthest from the equilibrium position.
3. Analyzing the forces:
At the extreme points of the motion, the bob is at its maximum displacement (a) from the equilibrium position. Therefore, the tension force (T) is at its maximum value.
4. Resolving the forces:
We can resolve the weight force (mg) into two components:
- The component parallel to the direction of displacement (a) is mg * (a/l).
- The component perpendicular to the direction of displacement is mg * (1 - a^2/l^2)^0.5.
5. Equilibrium condition:
At the extreme points of the motion, the bob is momentarily at rest. This means that the net force acting on the bob is zero.
6. Calculating the tension:
To find the maximum tension, we equate the sum of the components of the weight force and the tension force to zero.
7. Applying the equilibrium condition:
The net force in the vertical direction is zero, so we equate the sum of the components of the weight force and the tension force in the vertical direction to zero.
8. Mathematical calculation:
mg * (1 - a^2/l^2)^0.5 + T - mg = 0
9. Simplification:
T = mg - mg * (1 - a^2/l^2)^0.5
10. Further simplification:
T = mg * [1 - (1 - a^2/l^2)^0.5]
11. Final expression:
T = mg * [1 - (1 - a^2/l^2)]
T = mg * [1 - 1 + a^2/l^2]
T = mg * (a^2/l^2)
Therefore, the expression for the maximum tension in the string is mg * (a^2/l^2), which corresponds to option B.
1. Forces acting on the bob:
At the extreme points of its motion, the bob experiences two forces:
- The weight force (mg) acting vertically downward.
- The tension force (T) acting along the length of the string.
2. Maximum tension:
To find the maximum tension, we need to consider the point where the tension is maximum, which is at the extreme points of the motion, when the bob is farthest from the equilibrium position.
3. Analyzing the forces:
At the extreme points of the motion, the bob is at its maximum displacement (a) from the equilibrium position. Therefore, the tension force (T) is at its maximum value.
4. Resolving the forces:
We can resolve the weight force (mg) into two components:
- The component parallel to the direction of displacement (a) is mg * (a/l).
- The component perpendicular to the direction of displacement is mg * (1 - a^2/l^2)^0.5.
5. Equilibrium condition:
At the extreme points of the motion, the bob is momentarily at rest. This means that the net force acting on the bob is zero.
6. Calculating the tension:
To find the maximum tension, we equate the sum of the components of the weight force and the tension force to zero.
7. Applying the equilibrium condition:
The net force in the vertical direction is zero, so we equate the sum of the components of the weight force and the tension force in the vertical direction to zero.
8. Mathematical calculation:
mg * (1 - a^2/l^2)^0.5 + T - mg = 0
9. Simplification:
T = mg - mg * (1 - a^2/l^2)^0.5
10. Further simplification:
T = mg * [1 - (1 - a^2/l^2)^0.5]
11. Final expression:
T = mg * [1 - (1 - a^2/l^2)]
T = mg * [1 - 1 + a^2/l^2]
T = mg * (a^2/l^2)
Therefore, the expression for the maximum tension in the string is mg * (a^2/l^2), which corresponds to option B.
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A simple pendulum with length l and bob of mass m is executing S.H.M o...
The tension in the string would be maximum when bob will be passing mean position.
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