Centripetal Forces

Step 1: Concept
Centripetal acceleration .
Step 2: Analysis
Frequency , so . Radius . .
Step 3: Conclusion
.
Final Answer: (C)

Step 1: Expansion
. The expression is .
Step 2: Identifying Terms
Possible combinations to get : - coefficient = 1 - coefficient = - coefficient =
Step 3: Total Coefficient
Total = .
Final Answer: (b)

Step 1: Apply the concept of minimum velocity.
At the highest point of the vertical circle, the velocity should be such that the plane just completes the circular motion. The tension in the string is zero at this point. The only force acting on the plane is the gravitational force, and it provides the necessary centripetal force.

Step 2: Centripetal force at the highest point.
The centripetal force required to keep the plane in motion at the highest point is: where is the mass of the plane, is the velocity, and is the radius of the circle.

Step 3: Gravitational force.
The gravitational force acts downward and provides the required centripetal force. Thus, at the highest point, we have: Simplifying: Taking the square root: Thus, the minimum velocity at the highest point is .

Step 1: Understand the forces acting on the mass.
When the mass is whirled in a vertical circle, two forces act on it: - The tension in the wire ( ), - The gravitational force ( ) acting downward. The tension in the wire is greatest when the mass is at the lowest point of the circle. This is because, at the lowest point, the tension must support both the centripetal force (required to maintain circular motion) and the weight of the mass.

Step 2: Analyze the forces at different points.
- At the highest point of the circle, the gravitational force aids in providing the centripetal force. Hence, the tension is less than at the lowest point. - At the lowest point, the tension in the wire must overcome both the centripetal force and the gravitational force, leading to maximum tension and increasing the likelihood of the wire breaking.

Step 3: Conclusion.
Therefore, the wire is most likely to break when the mass is at the lowest point of the circle.

Step 1: Understanding the Concept:
At minimum deviation, and . Snell's law: .
Step 2: Detailed Explanation:
Given , . At minimum deviation, .
Snell's law:
.
Step 3: Final Answer:
Thus, angle of incidence = .

Step 1: Understanding the Concept:
Stefan-Boltzmann law: . Rate of emission .
Step 2: Detailed Explanation:
.
Step 3: Final Answer:
Thus, rate of emission increases by factor 16.

Concept: Rate of change of momentum = force.

Step 1: Centripetal force.


Step 2: Relation.

Concept:

Step 1: Maximum change in momentum.


Step 2: Minimum change in momentum.
Final Answer:

Concept: Angular velocity about a point: where is the distance from the point and is perpendicular component of velocity.

Step 1: Angular velocity about centre .


Step 2: Angular velocity about point .
Distance from A to particle varies, but for circular motion: (since diameter )

Step 3: Ratio.

Concept: For an inverse square central force: Potential energy: For circular motion:

Step 1: Find kinetic energy.


Step 2: Find potential energy.


Step 3: Total energy.
Putting :

Concept: When a vehicle moves on a flat circular road, friction provides the necessary centripetal force. Maximum frictional force: At maximum speed:

Step 1:Cancel common terms

Step 2:Solve for

Step 3:Substitute values

Step 4:Interpretation This is the maximum safe speed. Beyond this speed, friction will be insufficient and the car may skid outward. Final Answer : 30\,m/s