Question

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 _________ m/s.

Answer 1

Hint : The law of conservation of energy states that the total energy of an isolated system remains constant. We need to assign the sine value which is given by the slope of the inclined plane. Using that we can find the height of the inclined plane.

Formula used: The formulae used in the solution are given here.
 $ \tan \theta = \dfrac{{perpendicular}}{{base}} $ and $ \sin \theta = \dfrac{{perpendicular}}{{hypotenuse}} $ .

Complete step by step answer:
It is given that a solid sphere of mass 0.1 Kg and radius 2 cm rolls down an inclined plane 1.4 m in length.
Thus, the mass of the solid sphere is 0.1 kg and radius is 2 cm. Length of inclined plane is 1.4 m
Slope of a straight line is given by $ \tan \theta $ .
Given that the slope is 1 in 10.
Thus, $ \tan \theta = \dfrac{1}{{10}} $ .
 $ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{1}{{10}}} \right) $
Thus the value of $ \theta $ is 0.099.
Since $ \tan \theta = \dfrac{{perpendicular}}{{base}} $ and $ \sin \theta = \dfrac{{perpendicular}}{{hypotenuse}} $ ,
Therefore we have,
 $ \sin \theta = 0.099 $ .
The height of the inclined plane is given by, $ h' = h\sin \theta $ , where $ h = 1.4m $ in length and $ \sin \theta = 0.099 $ .
Putting the values into the formula,
 $ h' = 1.4 \times 0.099 $
 $ \Rightarrow h' = 0.14m $
According to the law of conservation of energy, energy can neither be created nor destroyed but can be transformed from one form to another.
We need to calculate the velocity. By using conservation of energy, we get,
 $ mgh' = \dfrac{1}{2}m{v^2} + \dfrac{1}{5}m{v^2} $
Assigning the values to this equation,
 $ 9.8 \times 0.14 = \dfrac{7}{{10}}{v^2} $
  $ \Rightarrow {v^2} = 1.96 $
Calculating the value of velocity,
 $ v = \sqrt {1.96} = 1.4m/s.\; $ .

Note:
In an isolated system such as the universe, if there is a loss of energy in some part of it, there must be a gain of an equal amount of energy in some other part of the universe. Although this principle cannot be proved, there is no known example of a violation of the law of conservation of energy.