The Question is from NEET guide , and it was created by the author ABDUL JILANI BASHA

Question

A bucket of water is swung in a vertical circle without spilling. What is the minimum speed the bucket must have at the top of the circle to prevent the water from falling out? (Let 'r' be the radius of the circle and 'g' be the acceleration due to gravity)

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Circular Motion and Centripetal Force
	**Related Topics**: Gravity, Minimum Speed for Vertical Circular Motion
	**Prerequisites**: Understanding of centripetal force, gravity as a force, circular motion
## SOLUTION PATHWAY
**Given Information**:
	* Radius of the circle: r
	* Acceleration due to gravity: g
	* Minimum speed to prevent water spillage at the top of the circle is required.
**Method & Steps**:
	1. Identify the forces acting on the water at the top of the circle.
		Reasoning: At the top, gravity acts downwards and the normal force from the bucket also acts downwards. The net force must provide the centripetal force for circular motion.
		Formula/Rule:  F_net =  F_gravity + F_normal = mv^2/r
	2. For the minimum speed, the normal force will be zero.
		Reasoning: If the normal force is zero, the water is just about to lose contact with the bucket.  Any slower, and the water would fall out.
		Key Point:  Minimum speed occurs when the normal force is zero.
	3. Solve for the minimum speed.
		Working: When F_normal = 0, then F_gravity = mv^2/r. Since F_gravity=mg, we have mg = mv^2/r.  Simplifying gives g = v^2/r, therefore v = sqrt(gr).
		Result: v = $\sqrt{gr}$
		Verification:  The units of $\sqrt{gr}$ are $\sqrt{m/s^2 * m}$ = $\sqrt{m^2/s^2}$ = m/s, which is a unit of speed.
 ## OPTION ANALYSIS
[A] :x: Incorrect. This would be the speed if the centripetal force was twice the gravity.
[B] ✓: Correct. This speed balances gravity and provides the necessary centripetal force for the water to stay in the circle.
[C] :x: Incorrect. This would be a lower speed than required.
[D] :x: Incorrect. This speed is higher than necessary and would result in a large normal force.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting that at the minimum speed, the normal force is zero. [How to avoid]: Focus on the condition where the water is just about to lose contact with the bucket.
		2.[Error 2]: Confusing the direction of centripetal force. [How to avoid]: The centripetal force always points towards the center of the circle.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that at minimum speed, the gravity alone is providing the centripetal force at the top of the circle.
		2.[Memory aid]:  Think of the water barely staying in the bucket; its speed is just enough to keep it moving in a circle.

ABDUL JILANI BASHA · Answer

$\sqrt{2gr}$

ABDUL JILANI BASHA · Answer

The correct Option is "$\sqrt{gr}$"

ABDUL JILANI BASHA · Answer

$\sqrt{gr/2}$

ABDUL JILANI BASHA · Answer

$2\sqrt{gr}$

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