A 1.00 -kg block is pushed up and down a rough plank of length L=2.00 m, inclined at 30.0^∘ abo

step 1 For this problem on the topic of energy conservation, we are told that a block with mass 1 kilog

A 1.00 -kg block is pushed up and down a rough plank of...

A 1.00 -kg block is pushed up and down a rough plank of length $L=2.00 \mathrm{~m},$ inclined at $30.0^{\circ}$ above the horizontal. From the bottom, it is pushed a distance $L / 2$ up the plank, then pushed back down a distance $L / 4$, and finally pushed back up the plank until it reaches the top end. If the coefficient of kinetic friction between the block and plank is 0.300 , determine the work done by the block against friction.

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Key Concepts

Work

Work is defined as the product of the force component along the displacement and the magnitude of the displacement. In situations where the force is variable or the path is complex, it is computed as the integral of the force component along the path, emphasizing that work is path dependent.

Kinetic Friction

Kinetic friction is the resistive force that opposes the motion of sliding objects. It is calculated using the coefficient of kinetic friction multiplied by the normal force. This force does negative work (dissipates energy) when an object moves, converting mechanical energy into thermal energy.

Normal Force on an Inclined Plane

On an inclined plane, the normal force is the component of the gravitational force perpendicular to the surface. It is given by the weight of the object multiplied by the cosine of the incline angle. This calculation is crucial since the normal force directly influences the magnitude of the frictional force.

Path Dependence of Friction Work

The work done against friction depends on the distance traveled along the surface. When an object follows a segmented or non-uniform path, the total work against friction is the sum of the frictional work over each segment, which requires careful accounting of the distance moved in each direction.

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