Calculating Work Done Against Gravity
The work done to move an object against a conservative force, like gravity, is equal to the change in its potential energy.
Potential Energy Formula
The gravitational potential energy ($U$) of a mass ($m$) at a distance ($r$) from the center of the Earth (mass $M$) is given by:
$U = -\frac{GMm}{r}$
Where $G$ is the gravitational constant.
Initial and Final States
- Initial State: At the Earth's surface, the distance from the center is $r_1 = R$. The initial potential energy ($U_1$) is:
$U_1 = -\frac{GMm}{R}$
- Final State: The mass is raised to a height $R$ above the surface. The final distance from the center is $r_2 = R + R = 2R$. The final potential energy ($U_2$) is:
$U_2 = -\frac{GMm}{2R}$
Work Done Calculation
The work done ($W$) is the change in potential energy:
$W = \Delta U = U_2 - U_1$
$W = \left(-\frac{GMm}{2R}\right) - \left(-\frac{GMm}{R}\right)$
$W = -\frac{GMm}{2R} + \frac{GMm}{R}$
$W = \frac{GMm}{R} \left(1 - \frac{1}{2}\right)$
$W = \frac{GMm}{2R}$
Relating to Surface Gravity
At the Earth's surface, the acceleration due to gravity is $g$. The force of gravity is $mg$. This force is also given by Newton's law of gravitation:
$mg = \frac{GMm}{R^2}$
From this, we can express $GM$ as:
$GM = gR^2$
Final Work Done Expression
Substitute $GM = gR^2$ into the work done equation:
$W = \frac{(gR^2)m}{2R}$
$W = \frac{mgR}{2}$
Therefore, the work done to raise the mass is $\frac{mgR}{2}$.
Similar Questions
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Important Questions from Mechanics
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
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The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
