Question
What is a conservative force? Prove that gravitational force is conservative, while frictional force is non-conservative.
✓
Answer
Conservative force : The work done against or by the force in moving a body depends only on initial and final positions of body and not on nature of path followed. For example, gravitational force is a conservative force.
Consider a body of mass 'm' to be moved to height ‘h’ over different paths from x to y.
In Mass is being raised vertically upwards along direction of force applied.
Work done = Force x distance
W1 = mg x h
In Mass 'm' is taken along a smooth inclined plane yz of height 'h' and inclination a.
Work done = F × yz
W2 = mg sin a × yz
$=\text{mg}\Big(\frac{\text{xy}}{\text{yz}}\Big)\times\text{yz}$
(F = mg sina as sine component contributes to applied force)
W2 = mgh
In Mass ‘m' is taken to height 'h' through a staircase having ‘n’ number of steps. Each step has height h' and width x' so work done in carrying the mass through horizontal distance $\text{x}'=\text{mg}\cos90^\circ=0$ and through vertical distance h' = mgh'.
Hence total work done in n steps is W3 = mgh' × n = mgh
In Mass ‘m’is carried to height-h’ in a zig-zag path. Zig-zag path can be divided into infinitesimally small steps of horizontal displacement dx and vertical displacement dh. As the work done along any horizontal displacement is zero. Total work done along all vertical paths will add up $\text{W}_4=\Sigma(\text{dh)}=\text{mgh}$
Thus $\text{W}_1=\text{W}_2=\text{W}_3=\text{W}_4=\text{mgh}$
$\therefore$ Whatever is the path followed to go from X to Y work done is the same. This shows that gravitational force is a conservative force.
Consider a body of mass 'm' to be moved to height ‘h’ over different paths from x to y.
In Mass is being raised vertically upwards along direction of force applied.
Work done = Force x distance
W1 = mg x h
In Mass 'm' is taken along a smooth inclined plane yz of height 'h' and inclination a.
Work done = F × yz
W2 = mg sin a × yz
$=\text{mg}\Big(\frac{\text{xy}}{\text{yz}}\Big)\times\text{yz}$
(F = mg sina as sine component contributes to applied force)
W2 = mgh
In Mass ‘m' is taken to height 'h' through a staircase having ‘n’ number of steps. Each step has height h' and width x' so work done in carrying the mass through horizontal distance $\text{x}'=\text{mg}\cos90^\circ=0$ and through vertical distance h' = mgh'.
Hence total work done in n steps is W3 = mgh' × n = mgh
In Mass ‘m’is carried to height-h’ in a zig-zag path. Zig-zag path can be divided into infinitesimally small steps of horizontal displacement dx and vertical displacement dh. As the work done along any horizontal displacement is zero. Total work done along all vertical paths will add up $\text{W}_4=\Sigma(\text{dh)}=\text{mgh}$
Thus $\text{W}_1=\text{W}_2=\text{W}_3=\text{W}_4=\text{mgh}$
$\therefore$ Whatever is the path followed to go from X to Y work done is the same. This shows that gravitational force is a conservative force.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Define linear momentum. Prove that for fixed mass, $P \propto v$ and for constant momentum $v \propto \frac{1}{m}$.
→Write Newton's formula for the speed of sound in air. Discuss the correction made by Laplace in this formula.
Figure shows a light spring balance connected to two blocks of mass 20kg each. The graduations in the balance measure the tension in the spring.
- What is the reading of the balance?
- Will the reading change if the balance is heavy, say 2.0kg?
- What will happen if the spring is light but the blocks have unequal masses?
Explain Doppler effect in sound. Obtain an expression for apparent frequency of sound when source and listener are approaching each other.
A cycle followed by an engine (made of one mole of perfect gas in a cylinder with a piston) is shown in Fig.
→
- In which part of the cycle heat is supplied to the engine from outside?
- In which part of the cycle heat is being given to the surrounding by the engine?
- What is the work done by the engine in one cycle? Write your answer in term of PA, PB, VA.
- What is the efficiency of the engine?[$\gamma=\frac{5}{3}$ for the gas], $\big(\text{C}_\text{v}=\frac{3}{2}\text{R for one mole}\big)$
Define conservative and non-conservative forces. Give example and properties of conservative forces.
A 5kg rifle fires a 58 bullet with a speed of 500ms-1. What kinetic energy is acquired.
A 5kg rifle fires a 58 bullet with a speed of 500ms-1. What kinetic energy is acquired.
- By the bullet and.
- By the rifle?
- Find the the ratio of the distance which the rifle moves backward while the bullet is in the barrel to the distance the bullet moves forward.
Prove that when the vector sum of all external forces acting on a system is zero, then the center of mass remains constant.
A body of mass 3kg is under a constant force, which causes a displacement S in metre in it, given by the relation $\text{S}=\frac{1}{2}\text{t}^2$, where t is in second. Find the work done by the force in 2s.
→A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period.
→$\text{T}=2\pi\sqrt{\frac{\text{m}}{\text{A}\rho\text{g}}}$
where m is mass of the body and ρ is density of the liquid.The energy of a silver atom with a vacancy in K shell is 25.31keV, in L shell is 3.56keV and in M shell is 0.530keV higher than the energy of the atom with no vacancy. Find the frequency of $\text{K}_\alpha\text{K}_\beta$ and $\text{L}_\alpha$ X-rays of silver.
→