A solid sphere of mass 0.1 kg and radius 2 cm rolls down an inclined plane 1.4 m in length (slope 1 in 10). Starting from rest its final velocity will be

The correct option is 1.4 m/sec.

Let's break down the physical process step-by-step to find the final velocity of the rolling sphere.

Step 1: Understand the given data
Mass of the solid sphere, $M=0.1\text{kg}$
Radius of the solid sphere, $R=2\text{cm}=0.02\text{m}$
Length of the inclined plane (distance traveled), $s=1.4\text{m}$
Slope of the inclined plane is 1 in 10, which means: $sin\theta =\frac{1}{10}=0.1$ (where $\theta$ is the angle of inclination)
Acceleration due to gravity, $g\approx 9.8{\text{m/s}}^2$
The sphere starts from rest, so its initial velocity $u=0$.

Step 2: Find the vertical height of the incline
The vertical height $h$ of the incline corresponding to the length $s$ is given by:
$h=s·sin\theta$
Substituting the values:
$h=1.4·0.1=0.14\text{m}$

Step 3: Apply the Law of Conservation of Energy
As the solid sphere rolls down without slipping, its potential energy at the top of the incline converts into both translational kinetic energy and rotational kinetic energy at the bottom.
$Mgh=\frac{1}{2}M{v}^2+\frac{1}{2}I{\omega }^2$

Where:
- $I$ is the moment of inertia of a solid sphere about its central axis, $I=\frac{2}{5}M{R}^2$.
- $\omega$ is the angular velocity, which is related to linear velocity $v$ by $\omega =\frac{v}{R}$ since it rolls without slipping.

Substituting $I$ and $\omega$ into the energy equation:
$Mgh=\frac{1}{2}M{v}^2+\frac{1}{2}\left(\frac{2}{5}M{R}^2\right){\left(\frac{v}{R}\right)}^2$
$Mgh=\frac{1}{2}M{v}^2+\frac{1}{5}M{v}^2$
Since $M$ is present in all terms, it cancels out:
$gh=\left(\frac{1}{2}+\frac{1}{5}\right)v^2$
$gh=\frac{7}{10}{v}^2$

Solving for $v$:
$v^2=\frac{10gh}{7}$
$v=\sqrt{\frac{10gh}{7}}$

Step 4: Calculate the final velocity
Substituting $g=9.8{\text{m/s}}^2$ and $h=0.14\text{m}$ into the formula:
$v=\sqrt{\frac{10·9.8·0.14}{7}}$
$v=\sqrt{\frac{13.72}{7}}$
$v=\sqrt{1.96}$
$v=1.4\text{m/sec}$

Thus, the final velocity of the sphere at the bottom of the incline is 1.4 m/sec.