The water skier in Figure 7-20 is at an angle of 32.5^∘ with respect to the center line of the b

step 1 In this problem, you have a boat pulling a skier. The diagram in the book looks something like t

The water skier in Figure 7-20 is at an angle of 32.5^∘ with...

The water skier in Figure $7-20$ is at an angle of $32.5^{\circ}$ with respect to the center line of the boat, and is being pulled at a constant speed of 15.6 $\mathrm{m} / \mathrm{s}$ . If the tension in the tow rope is $111 \mathrm{N},$ (a) how much work does the rope do on the skier in 10.0 $\mathrm{s} ?$ (b) How much work does the resistive force of water do on the skier in the same time?

Key Concepts

Work

Work is a measure of energy transfer when a force causes a displacement. It is calculated as the dot product of the force vector and the displacement vector, which means only the component of the force in the direction of displacement contributes. This fundamental concept in mechanics relates energy use to movement and is central to understanding how forces influence the kinetic energy of an object.

Force Decomposition

Force decomposition involves breaking a force into components along different directions, typically aligning one axis with the direction of movement. This is crucial when dealing with forces applied at an angle, as only the component of the force parallel to the displacement contributes to the work done. This concept is widely used in analyzing any system where forces are not applied along the standard coordinate axes.

Dot Product

The dot product is a mathematical operation used to calculate the work done by a force. It multiplies the magnitude of the force and the displacement and includes the cosine of the angle between them. This operation simplifies understanding how much of a force is effective in doing work, particularly when the force is applied at an angle.

Power

Power is the rate at which work is performed or energy is transferred. It quantifies how quickly work is done and is defined as the work done per unit of time. In many physical problems, understanding power helps in assessing the performance and efficiency of mechanical systems.

Resistive Forces

Resistive forces, such as friction or drag, oppose the motion of an object and consequently do negative work. In problems where an object moves at constant speed, the work done by the resistive force is equal in magnitude but opposite in sign to the work done by the applied forces. This concept is important for understanding energy dissipation in mechanical systems.

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