Question

An object of mass 1kg has a P.E. of 1J relative to the ground when it is at a height of: [Take \[g = 9.8\dfrac{m}{{{s^2}}}\]]
(A) 0.102
(B) 1
(C) 9.8
(D) 32

Answer 1

Hint: Use the potential energy formula $U = m.g.h$ considering the ground at zero potential. Here,
m= mass of the object
g= acceleration due to gravity
h= distance from the zero-potential level (generally ground)

Complete step by step answer:The amount of matter in a body is known as mass and is independent of gravity. This when multiplied with acceleration due to gravity becomes weight. Weight is nothing but the gravitational force exerted by the gravity on the body of mass m.
Therefore, by the definition of work done which is nothing but the dot product of force and displacement integrated over the end points of path (Since gravitational force field is a conservative field).
\[W = - \int\limits_{{x_1}}^{{x_2}} {F.dx = } - \int\limits_{{x_1}}^{{x_2}} {F\cos \theta dx} \]
Here force is the weight of the body and angle between the force and displacement is zero. Therefore,
\[\int\limits_{{x_1}}^{{x_2}} {F.dx = \int\limits_{{x_1}}^{{x_2}} {mg.dx = mg({x_2} - {x_1})} } \]
Let for simplicity ${x_2} = x;{x_1} = 0$. Here the negative sign indicates that the work is done by the gravitational field and the work is done on the body. So in a layman language, the gravitational field loses energy which is gained by the body and vice versa if the opposite route is taken. The potential energy is negative of work done (U=-W). Hence, the conservation of energy is upheld. So, we have,
\[
  U = - W \\
  U = m.g.x \\
 \]
Substituting the data from the question, we have,
\[
  U = mgx \Rightarrow x = \dfrac{U}{{mg}} \\
   \Rightarrow x = \dfrac{1}{{1 \times 9.8}}m \\
   \Rightarrow x = 0.102m \\
 \]
Therefore, at a distance of 0.102 meters, the body of mass 1kg has potential energy equal to 1J.
The correct answer is option A.

Note:Be careful in substituting data from the question in terms of units and dimensions. Here the question gave direct energy in joule. Sometimes it may occur that the energy is in calorie or the height is given in centi-meters.