Question

If a sphere is rolling on an inclined plane with a velocity v without slipping, the vertical height of the incline in terms of velocity will be:

A. \[\dfrac{{7v}}{{10g}}\]
B. \[\dfrac{{2{v^2}}}{{5g}}\]
C. \[\dfrac{{7{v^2}}}{{10g}}\]
D. \[\dfrac{{3v}}{{5g}}\]

Answer 1

Hint: Use the formula for velocity \[ = \sqrt {\dfrac{{2gh}}{{1 + \dfrac{I}{{M{R^2}}}}}} \]
I for solid sphere is \[\dfrac{2}{5}M{R^2}\]. Released from rest on an inclined smooth plane in a stationary fluid, a sphere accelerates along the plane under the influence of gravity and eventually reaches a terminal velocity. The variations of velocity with time and distance, the terminal velocity, the terminal distance (the practical distance required for a sphere from rest to its terminal velocity), are investigated through laboratory experiments and a theoretical analysis.

Complete step by step answer:
The relationship of the drag coefficient and the Reynolds number for the moving sphere with its terminal velocity is obtained and compared with that in the free fall. The effect of proximity of sidewalls of the flume on the fluid drag acting on the steady movement of the sphere is evaluated. The terminal velocity and the terminal distance against the sediment number are presented in dimensionless graphs.
Velocity of solid sphere on the bottom of inclined plane, \[v = \sqrt {\dfrac{{2gh}}{{1 + \dfrac{I}{{M{R^2}}}}}} \]
Where, g = gravitational acceleration, h = height, I = moment of inertia, M = mass, R = radius
Moment of inertia for solid sphere,
\[I = \dfrac{2}{5}M{R^2}\]
\[\therefore \] \[v = \sqrt {\dfrac{{2gh}}{{1 + \dfrac{I}{{M{R^2}}}}}} = \sqrt {\dfrac{{2 \times g \times h}}{{1 + \dfrac{{\dfrac{2}{5}M{R^2}}}{{M{R^2}}}}}} \]
\[ \Rightarrow \,v = \sqrt {\dfrac{{2gh}}{{1 + \dfrac{2}{5}}}} \]
\[ \Rightarrow \,v = \sqrt {\dfrac{{2gh}}{{\left( {1 + \dfrac{2}{5}} \right)}}} = \sqrt {\dfrac{{10}}{7}gh} \]
\[ \Rightarrow \] Squaring both sides,
\[{v^2} = \dfrac{{10}}{7}gh\]
\[ \Rightarrow \] \[h = \dfrac{{7{v^2}}}{{10g}}\]

So, the correct answer is “Option B”.

Note: The formula of velocity of solid sphere on the bottom of inclined plane and moment of inertia of sphere is required to solve these questions. The terminal velocity and the terminal distance against the sediment number are presented in dimensionless graphs. Given bed inclination as well as the properties of the fluid and the sphere, the terminal velocity and the terminal distance can be determined directly from the graphs. The experiments of the steady movement for a sphere rolling down a rough inclined boundary are also presented.