The force acting on a particle of mass m moving along the x-axis is given by F(x) = Ax² – Bx. Which one of the following is the potential energy of the particle ?
- (a)2Ax – B
- (b)– x²/6 (2Ax – 3B)
- (c)Ax³ – Bx²
- (d)Zero
Correct — B, – x²/6 (2Ax – 3B). For a conservative force in one dimension the force is minus the rate of change of potential energy with position, so the potential energy is minus the integral of the force. Integrating Ax² – Bx with respect to x gives Ax³/3 – Bx²/2, and putting the minus sign in front gives U(x) = –Ax³/3 + Bx²/2, taking the arbitrary constant as zero. Now expand the option and check. Multiplying out – x²/6 times (2Ax – 3B) gives –(2Ax³ – 3Bx²)/6, which is –2Ax³/6 + 3Bx²/6, that is –Ax³/3 + Bx²/2 — exactly the expression obtained by integrating. Option (b) is the same function written in factored form.
- (a)2Ax – B — This is the derivative of the given force, not its integral. Differentiating Ax² – Bx gives 2Ax – B, so this option goes one step in the wrong direction along the chain that links potential energy, force and its rate of change.
- (c)Ax³ – Bx² — This is close to an integral but wrong on two counts. The fractions 1/3 and 1/2 that integration produces have been dropped, and the overall minus sign that the definition demands is missing.
- (d)Zero — A potential energy of zero everywhere would mean no force at all, since the force is minus the slope of the potential energy. The stem gives a force that plainly varies with position, so the potential energy cannot be constant.
A force is conservative if the work it does between two points is independent of the path taken, and every such force can be written as minus the gradient of a potential energy function. In one dimension that reduces to F(x) equal to minus dU/dx, so U(x) is minus the integral of F(x) dx. The constant of integration is arbitrary and is fixed by choosing where U is to be zero, which is why only differences in potential energy have physical meaning.
The safe procedure with an item like this is to work forward from the option rather than only integrating the stem. Differentiate each candidate and see which one, with a minus sign in front of its derivative, reproduces the given force — that check is quick and immune to slips in the integration. Note also how the paper prints option (b): in the booklet it is a stacked fraction, x² over 6, multiplying the bracket, and it appears here flattened onto one line. Reading it as minus one-sixth of x² times (2Ax – 3B) is what the printed form means. Signs are the usual place marks are lost here — remember that a positive force pushes toward lower potential energy, which is where the minus sign comes from.
- For a conservative force in one dimension, F(x) = –dU/dx.
- Therefore U(x) is the negative of the integral of F(x) with respect to x.
- Integrating Ax² – Bx gives Ax³/3 – Bx²/2, so U(x) = –Ax³/3 + Bx²/2.
- The constant of integration is arbitrary, so only differences in potential energy are physically meaningful.
- Gravity and the spring force are conservative; friction is not, and has no potential energy function.
Option (b) is the integrated result written in factored form, which is why it looks unfamiliar at first sight.
- Forgetting the minus sign in F equals minus dU/dx.
- Differentiating the force when the question calls for integrating it.
- Dropping the fractions that integration of a power of x always produces.
NDA gives a force as a function of position and asks for the potential energy, or gives the potential energy and asks for the force, so the definition has to be usable in both directions.
The energy possessed by a body due to its change in position or shape is called
- (a) thermal energy
- (b) potential energy
- (c) kinetic energy
- (d) electric energy
Answer(b) potential energy
The definition this question makes you compute with — potential energy depends only on where the particle is, which is why it can be recovered from the force by integrating over position.
When a ball bounces off the ground, which of the following changes suddenly? (Assume no loss of energy to the floor)
- (a) Its speed
- (b) Its momentum
- (c) Its kinetic energy
- (d) Its potential energy
Answer(b) Its momentum
Another NDA mechanics item that turns on keeping scalars and vectors apart, the same care this question needs when the minus sign in the force-potential relation is applied.
- practice — not a real PYQ
If the potential energy of a particle is U(x) = kx²/2, the force acting on it is
- (a)–kx
- (b)+kx
- (c)kx²/3
- (d)zero
Answer(a) –kx — differentiating U and applying the minus sign gives the restoring force of a spring, which is Hooke's law.
- practice — not a real PYQ
Which of the following forces has no potential energy function associated with it?
- (a)Kinetic friction
- (b)Gravity
- (c)The elastic force of a spring
- (d)The electrostatic force
Answer(a) Kinetic friction — the work it does depends on the path taken, so it is non-conservative and no potential energy can be defined for it.