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
The power of a crane, which lifts a mass of $1000 \text{ kg}$ to a height of $20 \text{ m}$ in $10 \text{ s}$ is :
($g = 9.8 \text{ m/s}^2$)- The magnitude and direction of the acceleration produced in a body of mass $5 \text{ kg}$ when two mutually perpendicular forces $8 \text{ N}$ and $6 \text{ N}$ act on it, are respectively :
- A box of mass $15 \text{ kg}$ is kept on the floor of a stationary trolley. The coefficient of static friction between the box and the trolley is $0.12$. Keeping the box in stationary state over the trolley, the maximum acceleration with which the trolley can be moved horizontally is : ($g = 10 \text{ m/s}^2$)
A thin wire of length 'L' and linear mass density 'm' is bent into a circular ring (in x-y plane) with centre 'C' as shown in figure. The moment of inertia of the ring about an axis yy' (tangent in the plane) will be :
- When a ruler falls vertically, 5 different persons catch it with different reaction times. ($g = 9.8 \text{ m s}^{-2}$)
A. Person A has reaction time of $0.20 \text{ s}$.
B. Person B has reaction time of $0.22 \text{ s}$.
C. Person C has reaction time of $0.18 \text{ s}$.
D. Person D has reaction time of $0.19 \text{ s}$.
E. Person E has reaction time of $0.21 \text{ s}$.
What is the correct order of the distance travelled by the ruler for each person ? - The angular speed of a flywheel is increased from $600 \text{ rpm}$ to $1200 \text{ rpm}$ in $10 \text{ s}$. The number of revolutions completed by the flywheel during this time is :
A frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point $A \left( \theta = \frac{\pi}{2} \right)$ with identical uniform angular speeds in opposite directions, and meet again at point $B \left( \theta = -\frac{\pi}{2} \right)$. During this time, which of the following figures schematically represent the magnitude of the total linear momentum $\vec{P}$ of the system, as a function of $\theta$?
- Two planets $P_1$ and $P_2$ with equal mass have radii $R_1$ and $R_2$, respectively, where $R_2 = \frac{R_1}{2}$. The escape speeds of $P_1$ and $P_2$ are $v_1$ and $v_2$, respectively. Then $\frac{v_2}{v_1}$ is :
- A car travels on a circular racetrack of radius $50\text{ m}$, which is banked at an angle $\theta$. If the car travels at a speed $10\text{ ms}^{-1}$, then the wear and tear on its tyres is minimum. Taking the acceleration due to gravity to be $10\text{ ms}^{-2}$, the value of $\theta$ is :
- Consider a particle moving along a straight line, whose position as a function of time is given by $s(t) = \alpha t^2 - \beta t + \gamma$, where $\alpha = 1\text{ ms}^{-2}$, $\beta = 6\text{ ms}^{-1}$ and $\gamma = 5\text{ m}$. The average speed of the particle, in $\text{ms}^{-1}$, from $t = 0$ to $t = 6\text{ s}$ is :
Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
