Uniform Circular Motion

Answer (d)

Step 1: Understand the forces involved in motion on a banked curve.
When a vehicle travels on a banked road, the road is inclined at an angle to the horizontal. This banking is designed to help the vehicle navigate the turn. The primary forces acting on the truck are its weight ( ) acting vertically downwards and the normal reaction force ( ) exerted by the road, which acts perpendicular to the surface of the bank. For simplicity, to find the maximum permissible speed without relying on friction to prevent outward skidding, we first consider an ideal scenario where the frictional force between the tires and the road is negligible.
Step 2: Resolve the normal force into its horizontal and vertical components.
The normal force can be resolved into two components:
A vertical component acting upwards.
A horizontal component acting towards the center of the circular path.

Step 3: Apply Newton's second law in the vertical and horizontal directions.
Vertical Equilibrium: Since there is no vertical acceleration, the net vertical force must be zero. The upward vertical component of the normal force balances the weight of the truck: Horizontal Motion: The horizontal component of the normal force provides the necessary centripetal force required for the circular motion of the truck. The centripetal force is given by , where is the mass, is the speed, and is the radius of the circular path.
Step 4: Solve the system of equations for the maximum permissible speed .
To find , we can divide the equation for horizontal motion by the equation for vertical equilibrium: The normal force and the mass cancel out: Now, solve for :
Step 5: Substitute the given values to calculate the maximum permissible speed.
Given values are: Radius of curvature
Acceleration due to gravity
Banking angle

Substitute these values into the formula for : Therefore, the maximum permissible speed of the truck on this banked curve, without relying on friction to prevent outward skidding, is .

The centrifugal force (also called centripetal force) is given by: Substitute the given values , , and :
Thus, the centrifugal force is 1125 N.

Step 1: Identify the types of acceleration.
There are two types of acceleration here: \begin{itemize} \item Centripetal (radial) acceleration: \item Tangential acceleration (due to change in speed): \end{itemize}
Step 2: Find the net acceleration.
These accelerations are perpendicular to each other, so total acceleration is given by:

Speed is same, but after half revolution, direction reverses
Initial velocity = , Final =
Change in velocity =

Resolving forces on a banked curve, total normal force increases due to centripetal acceleration and friction. Approximate combined effect leads to:

Tension

The tension in the string provides the centripetal force, which is given by: where , , and .
Solving for :
Thus, the permissible maximum velocity is 5 m/s.

Given: Step 1: Calculate Acceleration
Use the first equation of motion: Step 2: Calculate Distance
Use the second equation of motion: Substitute the values: Thus, the distance covered is:

The body starts from rest ( ) with uniform acceleration . The velocity at seconds is , so: The displacement in the -th second is given by: For the -th second ( ): For the -th second ( ): Total displacement: Option (3) is correct. Options (1), (2), and (4) do not match.