Three bodies, a ring, a solid cylinder, and a solid sphere roll down the same inclined plane without slipping, they start from rest, and the radii and masses of the bodies are identical. Which body will reach the ground with maximum velocity?

Question

Question

Answer 1

option(2)

CONCEPT:

Whenever the block is rolling on a surface then there is no relative motion of the point of contact of the body with respect to the surface, Then this type of motion is called pure rolling motion or rolling motion.
The kinetic energy of a body rolling without slipping is the sum of Kinetic energies of :
- Translational Motion
- Rotational Motion

KE = Kinetic Energy of Translational Motion + Kinetic Energy of Rotational Motion.

KE = (1/2)MV²_cm+ (1/2)Iω²

Where M is the mass of the body, V is the velocity of the body, I is the moment of Inertia and ω is the angular velocity

EXPLANATION:

According to the Conservation of Energy:

Potential Energy (PE = m g h) lost by the body in rolling down the inclined plane is equal to K.E gained by the body

K.E gained by the body = Kinetic Energy of Translational Motion + Kinetic Energy of Rotational Motion.

K.E gained by the body = \(\frac{1}{2}MV^2cm + \frac{1}{2}Iw^2\)

Since v = ω R

If we can replace ω (angular velocity v²/ R² ) = \(\frac{1}{2}MV^2+ \frac{1}{2}MK^2.\frac{v^2}{R^2}\)

m g h = \(KE= \frac{1}{2}MV^2cm (1 +\frac{k^2}{R^2})\)

V = \(√{\frac{2gh}{1+ \frac{k^2} {r^2}}} \)

Clearly, the velocity V attained by the rolling body at the bottom of the inclined plane is independent of its mass.

For a ring:

K²= R² Where k is the radius of gyration and R is the radius of the ring.

V_Ring= \(√{\frac{2gh}{1+ \frac{k^2} {r^2}}} \) by putting the value of (R = 1cm) ( K = 1cm)

V_ring = √(gh)

For a solid cylinder:

k²= R²/ 2

V_cylinder = \(√{\frac{2gh}{1+ \frac{k^2} {r^2}}} \) = \(\sqrt {{\frac{4gh}{{3}}}}\)

For a solid sphere:

k²= 2R²/ 5

V_sphere= \(√{\frac{2gh}{1+\frac{2}{5}}}\) =\(\sqrt{{\frac{10gh}{7}}}\)

Clearly, among the three bodies, the sphere has the greatest and the ring has the least velocity of the center of mass at the bottom of the inclined plane.,