The Question is from neet-physics , and it was created by the author P.NARASIMHA

Question

A pendulum bob of mass $m$ swings in a vertical circle of radius $1.5$ m. What is the minimum speed at the bottom for it to complete the circle?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option B
## CONCEPT FRAMEWORK
 **Primary Concept**: Conservation of energy and centripetal force.
 **Related Topics**: Circular motion, potential energy, kinetic energy.
 **Prerequisites**: Understanding of energy conservation, centripetal force, and gravitational potential energy.
## SOLUTION PATHWAY
**Given Information**:
  Radius of the circle, $r = 1.5$ m
  Mass of the bob = $m$
  Acceleration due to gravity, $g = 9.81 m/s^2$ (assumed)
**Method & Steps**:
 1. At the top of the circle, the minimum speed is such that the centripetal force is equal to the weight of the bob.  This ensures the bob doesn't fall.
  Reasoning: At the top, the tension in the string is zero (minimum speed). The centripetal force is provided solely by gravity.
  Formula/Rule: $mg = \frac{mv_{top}^2}{r}$ 
 2. Apply conservation of energy between the bottom and top of the circle. The total energy (kinetic + potential) remains constant.
  Working: At the bottom, the kinetic energy is $\frac{1}{2}mv_{bottom}^2$ and the potential energy is 0 (taking the bottom as the reference point). At the top, the kinetic energy is $\frac{1}{2}mv_{top}^2$ and the potential energy is $2mgr$.
  $ \frac{1}{2}mv_{bottom}^2 = \frac{1}{2}mv_{top}^2 + 2mgr $
 3. Substitute $v_{top}^2 = gr$ (from step 1) into the energy equation and solve for $v_{bottom}$.
  $ \frac{1}{2}mv_{bottom}^2 = \frac{1}{2}m(gr) + 2mgr = \frac{5}{2}mgr $
  $ v_{bottom}^2 = 5gr $
  $ v_{bottom} = \sqrt{5gr} = \sqrt{5 	imes 9.81 	imes 1.5} = \sqrt{73.575} \approx 8.57 m/s $
  Result: $v_{bottom} \approx 8.57 m/s$
  Verification: Check the units – they are consistent (m/s). The magnitude seems reasonable for this type of problem.
 ## OPTION ANALYSIS
A] :x: Incorrect calculation or incorrect application of energy conservation.
B] ✓: Correct application of energy conservation and centripetal force principles.
C] :x: Incorrect calculation or missing a factor in the energy equation.
D] :x: Incorrect calculation or an incorrect assumption about the forces involved.
 ## LEARNING AIDS
 **Common Mistakes**:
  1. [Forgetting potential energy]: Not considering the change in potential energy between the top and bottom points leads to an incorrect answer.
  2. [Incorrectly applying centripetal force]:  Not understanding that at the minimum speed at the top, the centripetal force is solely provided by gravity.
 **Quick Tips**:
  1. [Energy conservation shortcut]:  Directly use the energy conservation equation to relate the speeds at the top and bottom, avoiding separate calculations for each point.
  2. [Visualize the motion]: Draw a diagram of the pendulum bob's motion to help visualize the forces and energy changes.

P.NARASIMHA · Answer

3.83 m/s

P.NARASIMHA · Answer

The correct Option is "8.57 m/s"

P.NARASIMHA · Answer

5.42 m/s

P.NARASIMHA · Answer

6.64 m/s

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