You are rearranging the furniture in your living room. In...

You are rearranging the furniture in your living room. In doing so, you push a $30 \mathrm{~kg}$ sofa to the left a total distance of $2 \mathrm{~m}$ and then you push it back to the right a distance of $1 \mathrm{~m}$. The coefficient of kinetic friction between the sofa and the hardwood floor is 0.2 . (a) What is the work done by friction in moving the sofa? (b) Suppose that you had instead simply moved the sofa to its final position by moving it directly $1 \mathrm{~m}$ to the left. What is the work done by friction in this case? (c) From the results you obtained in parts (a) and (b), what can you conclude about the nature of the friction force? Is it a conservative force? Explain.

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Key Concepts

Work

Work is defined as the dot product of the force vector and the displacement vector, representing the energy transferred by a force acting over a distance. It involves both the magnitude of the force and the component of the displacement in the direction of the force, yielding a positive value when force and displacement are in the same direction, and a negative value when they oppose each other.

Kinetic Friction

Kinetic friction is the resistive force that opposes the motion of objects sliding against each other, calculated as the product of the coefficient of kinetic friction and the normal force. It performs negative work during motion because it acts in the direction opposite to the object's displacement, converting mechanical energy into thermal energy.

Conservative Forces vs. Nonconservative Forces

Conservative forces, such as gravity, have the property that the work done by these forces is independent of the path taken and can be described by a potential energy function. In contrast, nonconservative forces like friction depend on the path, meaning the work done by friction varies with the distance traveled, and energy is dissipated rather than stored as potential energy.

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