Work By A Position-dependent Force

Question

A single force directed along the x-axis varies with position as F = 3x^2 newtons and acts on a 1 kg particle initially at rest at the origin. What speed does the particle have after moving to x = 2 m?

RankGuru · Accepted Answer

A force that depends on position requires integration to find its work, $W = \int_{0}^{2} F\,dx = \int_{0}^{2} 3x^2\,dx$. Evaluating the integral gives $W = [x^3]_{0}^{2} = 2^3 - 0 = 8	ext{ J}$. Since this is the only force, the work-energy theorem equates it to the gained kinetic energy: $8 = 	frac{1}{2}(1)v^2$, so $v^2 = 16$ and $v = 4	ext{ m/s}$. The option 8 m/s mistakes the work value in joules for the speed. The option 16 m/s reports $v^2$ rather than $v$. The option 2 m/s arises from forgetting to integrate and using $F$ at a single point. The need for integration arises because the force is not constant over the displacement; multiplying its value at any single point by the distance would systematically misrepresent the true accumulated work. Geometrically, the integral of $3x^2$ corresponds to the steadily growing area beneath a parabolic force curve, which weights later positions far more heavily than the early ones near the origin. This demonstrates the NCERT method of combining integration of a variable force with the work-energy theorem. As a check, the units of $\int 3x^2 dx$ are $	ext{N}\cdot	ext{m}$, that is joules, and $	frac{1}{2}(1)(4)^2 = 8	ext{ J}$ recovers the computed work.

Work By A Position-dependent Force

Select the correct option:

🔒 Solution Hidden from View

More work, energy and power Practice Questions

A rubber ball is dropped from a height of 2 m onto a hard floor and rebounds to a height of 1.28 m a...

A satellite moves in a perfectly circular orbit around the Earth under the gravitational pull direct...

A cyclist of total mass 70 kg rides up a slope inclined at 30 degrees to the horizontal at a steady ...

A small stone tied to a string is whirled in a vertical circle of radius 2 m. What is the minimum sp...

Two bodies of different masses possess the same magnitude of linear momentum while moving in a strai...

A ball is thrown vertically upward and returns to the same height from which it was launched on a wi...

A stone of mass 0.5 kg is dropped from rest at a height of 8 m above the ground. Ignoring air resist...

A weightlifter raises a barbell of mass 80 kg from the floor to a height of 2 m above the floor in 4...

A 2 kg lump of clay moving at 6 m/s strikes and sticks to a 4 kg block initially at rest on a fricti...

A car engine delivers a constant power of 50 kW while the car moves at a steady speed of 25 m/s alon...

A wooden crate of mass 10 kg is dragged 5 m up a rough incline by a force, and friction exerts a con...

Which of the following statements correctly distinguishes a conservative force from a non-conservati...

About This Question

Solution