A thick walled hollow sphere has outer radius R. It rolls down an inclined plain Without slipping and its speed at the bottom is v. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom will be 5v/4. What is radius of gyration of the sphere.

Open in App

Solution

Initially, when the sphere is rolling down the include without any slipping, then let the angular velocity of the sphere at the bottom of the incline be w.

No slipping condition, V = wR
Furthermore, due to conservation of energy
PE lost = KE gained
mgh = 1/2mv^2 + 1/2 Iw^2 (Where h is the height of the incline and I is the moment of inertia of sphere around its center)
Also, I = mRg ^2 where Rg is the radius of gyration and also v = wR (No slipping condition)
Hence, mgh = 1/2mv^2 + 1/2 m*Rg ^2 v^2 /R^2
2gh = [1+(Rg/R)^2]v^2 (1)

Furthermore, under no rolling case
mgh = 1/2m(5v/4)^2
v^2 = 32/25 gh
Putting it in the equation 1
2gh = [1+(Rg/R)^2]32/25 gh
Hence, Rg = 3/4R
So, the radius of gyration of the sphere around its center is 3/4 R

Suggest Corrections

Similar questions

Q. A thick walled hollow sphere has outer radius

R

. It rolls down an inclined plane without slipping and its speed at the bottom is

v

. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom will be

5 v / 4

. What is the radius of gyration of the sphere?

Q. A thick walled hollow sphere has outer radius

R

. It rolls down an inclined plane without slipping and its speed at the bottom is

v

. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom is

5 v / 4

. What is the radius of gyration of the sphere?

Q. A thick walled hollow sphere has outer radius

R

. It rolls down an inclined plane without slipping and its speed at bottom is

v_{0}

. Now the incline is waxed so that the friction becomes zero. The sphere is observed to slide down without rolling and the speed now is

(5 \frac{v_{0}}{4})

. The radius of gyration of the hollow sphere about the axis through its centre is

Three similar spheres of mass m and radius r are moving down along three inclined planes A, B and C of similar dimensions. Sphere on inclined plane A rolls down in pure rolling, on B it partially rolls and partially slides down the plane whereas on plane C it slides down without rolling, then at the bottom of planes

Q. A solid sphere rolls down an inclined plane and its velocity at the bottom is

v_{1}

. Then same sphere slides down the plane (without friction) and let its velocity at the bottom be

v_{2}

. Which of the following relation is correct

Join BYJU'S Learning Program