Uniform Circular Motion

Step 1: Condition for water not falling down.
To prevent the water from falling, the centripetal force must be at least equal to the weight of the water. The condition for the critical velocity is:

Step 2: Relating velocity and frequency.
The velocity for a body in circular motion is related to frequency by:

Step 3: Solving for minimum frequency.
Equating the two expressions for :

Step 4: Conclusion.
The minimum frequency required is .

Step 1: Tension in the wire.
The tension in the wire when the mass is at the lowest point of the circle is the highest because, at that point, both the gravitational force and the centripetal force act in the same direction, requiring the maximum tension to hold the mass. At the highest point of the circle, the forces work against each other, so the tension is less.

Step 2: Analyzing the options.
- (A) The wire is horizontal, but the tension will not be at its maximum at this point. - (B) The mass is at the lowest point of the circle, where the chances of breaking the wire are maximum because the tension is highest. - (C) The wire makes an angle of 60° with the horizontal; this is not the point of maximum tension. - (D) The mass is at the highest point of the circle, but the tension here is less than at the lowest point.

Step 3: Conclusion.
The correct answer is (B), as the chances of breaking the wire are maximum when the mass is at the lowest point of the circle.

Step 1: Relationship between linear and angular velocity.
The linear velocity is related to the angular velocity by the formula: where is the radius of the circular path and is the angular velocity.
Step 2: Applying the given conditions.
Since both cars complete one revolution in the same time , their angular velocities are related by: Since the time is the same for both, the linear velocities of the cars are:
Step 3: Ratio of linear speeds.
The ratio of linear speeds is:
Step 4: Conclusion.
Thus, the ratio of the linear speeds is . Therefore, the correct answer is (A).

Step 1: Understanding the problem.
The ball is moving in a horizontal circular path, and the forces acting on it are tension in the string and the centripetal force required for circular motion.

Step 2: Using the centripetal force formula.
The centripetal force required to maintain circular motion is given by , where is the angular velocity and is the length of the string. The tension provides the centripetal force. Thus, .

Step 3: Solving for angular velocity.
Solving for , we get: Thus, the angular velocity of the ball is .

Step 4: Conclusion.
The correct answer is (A).

Step 1: Identify the force acting on the mass.
When a mass attached to a spring is whirled in a horizontal circle, the required centripetal force is provided by the spring force. Thus, where is the spring constant, is the elongation, is the angular speed, and is the radius of the circular motion.

Step 2: Write equations for the two given conditions.
Let the natural length of the spring be .
For angular speed , elongation : When the angular speed is doubled, , elongation :

Step 3: Eliminate constants and solve for the natural length.
Dividing the second equation by the first:

Step 4: Final conclusion.
The original (natural) length of the spring is .

Step 1: Write the formula for centripetal force.

Step 2: Write force equations for both stones.
For lighter stone: For heavier stone:
Step 3: Equate the forces.

Step 4: Simplify.

Step 5: Conclusion.
The tangential speed of the lighter stone is 3 times that of the heavier stone.

Step 1: Determine angular displacement of minute hand.
The minute hand completes one full revolution in minutes, i.e. in seconds.
Step 2: Calculate angular speed.

Step 3: Conclusion.
The angular speed of the minute hand is .

Step 1: Understanding the forces involved.
In this problem, the particle rotates in a horizontal circle inside a conical funnel. The centripetal force is provided by the component of the gravitational force acting along the surface of the funnel. This force is balanced by the vertical component of the tension in the string. The height of the circle from the vertex of the funnel can be related to the velocity and the radius using the equation: where is the speed of the particle and is the acceleration due to gravity.
Step 2: Conclusion.
Thus, the height of the circle from the vertex of the funnel is , which is option (C).

Step 1: Understanding the motion of the particle.
As the particle moves along a circular path with decreasing speed, it experiences a change in velocity. The direction of the angular momentum of the particle (which is perpendicular to the plane of motion) remains unchanged. The decrease in speed implies that the particle is losing kinetic energy, but the direction of the angular momentum vector remains constant.
Step 2: Conclusion.
Thus, the correct answer is (C) the direction of angular momentum remains constant.

Step 1: Forces acting on the vehicle.
For a vehicle moving on a banked curve, the forces acting are: Gravitational force ( ). Normal reaction force ( ). Frictional force ( ). The net force provides the necessary centripetal force for circular motion. Step 2: Resolving forces.
The component of the normal reaction along the radius provides the centripetal force: where is the speed, is the radius of the curve, and is the banking angle. Step 3: Frictionless case (Safety speed).
For the safety speed, assume friction is negligible. The normal force's component balances the centripetal force: Solve for : Step 4: Final Answer.
The safety speed is:

Step 1: Relationship between centripetal force and linear momentum.
The centripetal force for a particle in uniform circular motion is given by: where is the mass of the particle, is its velocity, and is the radius of the circular path. The magnitude of linear momentum is given by: Step 2: Ratio of to .
The ratio of centripetal force to linear momentum is: Step 3: Conclusion.
The ratio of centripetal force to linear momentum is .

A car of mass 1000 kg is moving in a circular path of radius 50 m with a speed of 20 m/s. We need to calculate the centripetal force acting on the car. The formula for centripetal force is:

Where:

Plugging in the values, we get:

Simplifying further:

The correct computation yields:

Thus, the centripetal force acting on the car is .

Concept:
For capacitor in AC: Step 1: Find and .
Step 2: Calculate capacitive reactance.
Step 3: Find current.
Step 4: Conclusion. Ammeter reading =

Concept:
For an ideal gas: Step 1: Express in terms of .
Step 2: Substitute in Mayer's relation.
Step 3: Find .
Step 4: Find .
Step 5: Conclusion.

Step 1: Definition
Optical path .
Step 2: Comparison
Without film, path length .
With film, optical path .
Step 3: Difference
Increase .
Final Answer: (D)

Step 1: Zener Operation
A Zener diode works as a voltage regulator only in the **reverse breakdown region**.
Step 2: Identifying Regions
- 'ab' and 'bc' are in the forward bias region. - 'cd' is the reverse bias region before breakdown (leakage current). - 'de' is the breakdown region where voltage remains almost constant despite large changes in current.
Step 3: Conclusion
Region 'de' is the regulation region.
Final Answer: (D)

The centripetal force acting on an object moving in a circle is given by: Where: - is the mass of the car, - is the speed of the car, - is the radius of the circular path. Now, substitute the values: Thus, the centripetal force acting on the car is .

At the topmost point of the vertical circular motion, the only forces acting on the object are the tension in the string and the gravitational force. To ensure the string remains taut, the centripetal force must be at least equal to the weight of the object. The centripetal force is provided by the tension in the string and gravity. Let be the tension and the weight of the object. For the string to remain taut, the condition at the topmost point is: At the minimum speed, , so the equation becomes: Solving for , we get: Thus, the minimum speed required at the topmost point is .

Concept: When a vehicle moves along a curved road, it requires a centripetal force directed towards the center of the circular path. On a banked road, the road is inclined at an angle with respect to the horizontal. This inclination helps provide the necessary centripetal force through the components of the normal reaction. A banked road allows vehicles to move safely around curves even without relying entirely on friction. The horizontal component of the normal reaction provides the centripetal force required for circular motion.

Step 1: Forces acting on the vehicle.
Two main forces act on the vehicle:
• Weight of the vehicle acting vertically downward
• Normal reaction perpendicular to the road surface

Step 2: Resolving forces.
The components of the normal reaction are:
• balancing the weight
• providing the centripetal force

Step 3: Applying circular motion condition.
Centripetal force required: Equating the horizontal component: From vertical balance: Dividing the two equations: Therefore, Thus, the safe speed of a vehicle on a banked curve is given by:

Concept: When a vehicle moves along a curved banked road, the road is inclined at an angle . This inclination helps provide the necessary centripetal force required for circular motion without relying on friction. For a vehicle moving safely without slipping, the required centripetal force is provided by the horizontal component of the normal reaction.

Step 1: Forces acting on the vehicle. Two main forces act on the vehicle:
• Weight acting vertically downward
• Normal reaction from the road surface Resolving the normal reaction:

Step 2: Dividing the equations.

Step 3: Solving for velocity. Thus, the safety speed on a banked road is given by