The Question is from neet-physics work-energy-power vertical-circular-motion, and it was created by the author Mr. ANIL KUMAR.G

Question

A small bob of mass 'm' attached to a string of length 'l' is undergoing vertical circular motion. At the highest point of the circle, the tension in the string is zero. If the bob was given an initial velocity 'u' at the bottommost point, which of the following expressions correctly represents 'u'?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option C
## CONCEPT FRAMEWORK
	**Primary Concept**: Conservation of Energy and Circular Motion
	**Related Topics**: Potential Energy, Kinetic Energy, Centripetal Force
	**Prerequisites**: Understanding of energy conservation, circular motion, and forces.
## SOLUTION PATHWAY
**Given Information**:
	* Mass of the bob: 'm'
	* Length of the string: 'l'
	* Tension at the highest point: 0
	* Initial velocity at the bottommost point: 'u'
**Method & Steps**:
	1.  **Analyze forces at the highest point:** At the highest point, the only force acting on the bob is gravity (mg) since tension is zero. This force provides the necessary centripetal force for circular motion.
		Reasoning: For circular motion, the net force towards the center is the centripetal force.
		Formula/Rule:  $F_c = \frac{mv^2}{r}$, where $F_c$ is the centripetal force, 'm' is the mass, 'v' is the velocity, and 'r' is the radius.
	2.  **Calculate velocity at the highest point:** Since tension is zero at the highest point,  $mg = \frac{mv_t^2}{l}$ , where $v_t$ is the velocity at the top.
		Working: $v_t^2 = gl$, so $v_t = \sqrt{gl}$.
		Key Point: This is the minimum velocity required at the top to complete the circle.
	3.  **Apply conservation of energy:** The total energy at the bottom (kinetic energy) must equal the total energy at the top (kinetic + potential energy).
		Reasoning: Energy is conserved in the absence of non-conservative forces like friction.
		Formula/Rule: $KE_i + PE_i = KE_f + PE_f$ where KE is kinetic energy, PE is potential energy, i is initial, and f is final.
	4.  **Calculate initial velocity at the bottom:**
		Working: $\frac{1}{2}mu^2 + 0 = \frac{1}{2}mv_t^2 + mg(2l)$. Substituting $v_t^2=gl$, we get  $\frac{1}{2}mu^2 = \frac{1}{2}mgl + 2mgl$. Simplifying, $\frac{1}{2}mu^2 = \frac{5}{2}mgl$, and $u^2 = 5gl$. Therefore, $u = \sqrt{5gl}$.
		Result: Initial velocity at the bottom, $u = \sqrt{5gl}$.
		Verification: The units match for velocity, and the result makes sense in the context of circular motion.
 ## OPTION ANALYSIS
[A] :x: $\sqrt{gl}$ is the velocity at the highest point, not the initial velocity at the bottom.
[B] :x: $\sqrt{3gl}$ is incorrect for this scenario.
[C] ✓:  $\sqrt{5gl}$ is the correct initial velocity required for the bob to just complete the circle.
[D] :x: $\sqrt{7gl}$ is incorrect for this scenario.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.  **Confusing velocity at top and bottom**:  Make sure to clearly identify which point you are considering.
		2.  **Forgetting potential energy difference**:  Remember the change in potential energy is $mg(2l)$ when going from the bottom to the top of the circle.
	**Quick Tips**:
		1.  **Minimum velocity at top**:  For vertical circular motion, the minimum velocity at the top is $\sqrt{gl}$ when tension is zero.
		2.  **Energy Conservation**:  Always apply energy conservation when dealing with motion under gravity.

Mr. ANIL KUMAR.G · Answer

$\sqrt{gl}$

Mr. ANIL KUMAR.G · Answer

$\sqrt{3gl}$

Mr. ANIL KUMAR.G · Answer

The correct Option is "$\sqrt{5gl}$"

Mr. ANIL KUMAR.G · Answer

$\sqrt{7gl}$

NEET Physics Work, Energy, and Power Vertical Circular Motion MCQs

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