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
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 :
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 : ($g = 10 \text{ m/s}^2$)
The following plots show variation of velocity ($v$) with time ($t$) of a ball thrown vertically upward, and falling back. Which of the following plots is/are correct?
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 :
Important Questions from Mechanics
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
In the given figure the blocks $A$, $B$ and $C$ weigh 4 kg, 6 kg and 8 kg respectively. The co-efficient of sliding friction between any two surfaces is 0.5. The force $\vec{F}$ required to slide the block $C$ with constant speed is ______ N. (Use $g = 10 \text{ m/s}^2$)
- The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5/x$. The value of $x$ is _______.
- In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division $= 0.05 \text{ mm}$, then the least count of the vernier callipers is _______ mm.
- A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^\circ$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _______.
