Rotational Motion

A car travels around a banked curve where the radius is 48 m and the banking angle is 15°. To find the maximum speed at which the car can travel without skidding on a slippery surface, we must consider the conditions of equilibrium and the centripetal force required for circular motion.

The banking of the road provides a component of the normal force that acts as the centripetal force. The component of gravitational force parallel to the road acts downward along the slope.

Given data:

• Radius
• Banking angle

The car does not skid if the gravitational component along the slope is balanced by the lateral component of the normal force along the direction of the centripetal force. Since there is no skidding even on a slippery surface, we ignore friction, and the centripetal force is provided entirely by the component of the normal force:

Substituting the given values,

Calculate :

Finding , the speed in m/s:

To convert this speed to kilometers per hour, use the conversion factor:

Thus, the closest answer to our calculation as per given options, considering any rounding differences in an actual test scenario, is 30.6 km/h.

To find the angular acceleration , we use the relation between torque , moment of inertia , and angular acceleration:

The torque is given by the product of the applied force and the radius of the disk:

Substitute the given values: , , and .

Calculate the torque:

Now, substitute into the first equation to solve for :

Solve for :

The angular acceleration is 75 rad/s .

A uniform hollow cylinder is a cylindrical shell characterized by having both an inner and an outer radius. To find the moment of inertia of the hollow cylinder about its axis, we integrate across the volume of the cylindrical shell. The moment of inertia for a differential mass element at a distance from the axis is given by .

The volume of the hollow cylinder is determined by subtracting the volume of the inner cylinder from the volume of the outer cylinder:

Given the density , the mass of the hollow cylinder can be expressed as:

For a small ring element at radius , the differential area is the area of the ring: , and thus the differential volume of the ring .

The differential mass is:

To find the moment of inertia of the entire cylinder, integrate over the thickness of the shell from to :

The definite integral evaluates to:

Plug this result back into the expression for :

Simplifying gives:

Substituting , which implies , we can write:

Notice that:

(by polynomial identity)

So, finally:

The correct answer is therefore .