Q.Obtain an expression for the potential energy of a spring having spring constant k when it is extended through a length $x$.

Work done in stretching a spring FROM one extension TO another (not from zero) is W = (1/2) k (x2^2 - x1^2), since spring PE = (1/2)kx^2 at each extension.

Spring potential energy at extension x: U(x) = (1/2) k x^2

Work done in extending from x1 = 5 cm = 0.05 m to x2 = 15 cm = 0.15 m equals the change in spring PE:

W = U(x2) - U(x1) = (1/2) k (x2^2 - x1^2)

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For a conservative force, the work done around any closed path is always zero.

A force is conservative if the work it does on a particle moving between two points is independent of the path taken, and depends only on the initial and final positions. For a closed path, the initial and final positions are the same point, so the net work done, W = -(change in potential energy) = -(U_final - U_initial) = 0, since U_final = U_ …

U = ½kx^2 with the same x gives U_A/U_B = k1/k2. Answer (D).

The potential energy stored in a stretched spring is U = ½ k x^2, where x is the extension.

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Friction is non-conservative; the other three are conservative. Answer (A).

A conservative force is one for which the work done depends only on the end points (and is zero around any closed loop), so energy can be stored as potential energy.

• Electrostatic, gravitational and spring forces are all conservative (each has an associated potential energy). …

Walking requires a horizontal push from friction between the feet and the ground; on a perfectly smooth surface, the only force the ground can exert is the normal reaction, which is always perpendicular to the surface (and hence to any displacement along it).

The normal reaction $\vec N$ is perpendicular to the instantaneous displacement $d\vec s$ along the surface at every point of the path, so the elemental work $dW=\vec N\cdot d\vec s = 0$ throughout the walk. There is no friction force (surface is smooth) to do any tangential work either. Since g …

For a conservative force, W = −ΔU, so positive work means potential energy falls.

For any conservative force, the work done by the force on a body is related to the change in potential energy by:

$W = -\Delta U = -(U_f - U_i)$

If the conservative force does positive work (W > 0), then:

$U_f - U_i < 0 \quad \Rightarrow \quad U_f < U_i$

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For any conservative force, the work done around a closed path is always zero, because the potential energy at the start and end points is the same.

A force F is conservative if the work it does on a body moving between two points is independent of the path taken, and depends only on the initial and final positions.

Such a force can be written as the negative gradient of a potential energy function U: F = -dU/dx (in one dimension).

Work done W = -(Uf - Ui).

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A conservative force does path-independent work and conserves mechanical energy; friction does neither.

A force is conservative if the work it does between two points is independent of the path taken (equivalently, work done over a closed loop is zero), and the energy "lost" to it can always be recovered as potential energy — e.g. gravity, spring force.

Friction, however:

• Always opposes relative motion and does negative work on the moving body. …

A conservative force is one whose work is path-independent and whose work over a closed loop is exactly zero.

A force is called conservative if it satisfies special properties that let us define a potential energy for it. The two defining (and most commonly asked) properties are:

1. Path independence: The work done by a conservative force in moving a body from one point to another depends ONLY on the initial and final positions of the body, and not at all on the path taken between them. For example, the work done by gravity in lifting a mass from the ground to a height h is mgh, regardless of whether the path taken is straight up, a ramp, or a curved staircase.

2. Zero work over a closed path: If a body starts and ends at the SAME point (a closed loop), the total work done by a conservative force over that round trip is zero. This follows directly from path independence, since the 'initial' and 'final' positions are identical.

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Gravity and spring force are classic conservative forces (work done is path-independent, zero over a closed loop); friction and air resistance are classic non-conservative forces (work done depends on the path taken and dissipates mechanical energy as heat).

A conservative force is one for which the work done in moving a body between two points does not depend on the path taken, and the work done over any closed path is zero. Such forces can be derived from a potential energy function.

• Example: Gravitational force — the work done by gravity depends only on the change in height, not the path; a potential energy (mgh) can be defined for it.
• Another example: the spring (restoring) force, F = -kx, with potential energy (1/2)kx^2. …