The force component along the displacement varies with the...

The force component along the displacement varies with the magnitude of the displacement, as shown in the graph. Find the work done by the force in the interval from (a) 0 to $1.0 \mathrm{m},$ (b) 1.0 to $2.0 \mathrm{m},$ and (c) 2.0 to $4.0 \mathrm{m}$ (Note: In the last interval the force component is negative, so the work is negative.

Key Concepts

Work Done by a Force

Work is defined as the energy transferred when a force acts on an object over a displacement. It is calculated by taking the dot product of the force vector and the displacement vector, and it indicates whether energy is being added to or removed from an object.

Variable Force

In many real-world problems, the force acting on an object is not constant but varies with position, time, or some other factor. When force varies with displacement, determining the work done requires careful consideration of how the force changes over the distance the object moves.

Definite Integration

Definite integration is a mathematical tool used to calculate the total accumulation of a quantity when it varies continuously over an interval. In the context of variable force, it is used to sum up the contributions of an infinitesimally small work over the displacement to obtain the total work done.

Area Under the Force-Displacement Curve

The work done by a variable force can be visually and mathematically represented as the area under the curve in a force-displacement graph. Positive areas contribute positive work, while negative areas indicate that the force is doing negative work, meaning it opposes the displacement.

Sign Convention in Work Calculations

The sign of work is important in determining whether energy is being transferred to or from the system. When a force component is in the same direction as the displacement, the work done is positive, and when the force opposes the displacement, the work done is negative.

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