Question

A tow rope, parallel to the water, pulls a water skier directly behind a boat with constant velocity for a distance of \(65 \mathrm{~m}\) before the skier falls. The tension in the rope is \(120 \mathrm{~N}\). (a) Is the work done on the skier by the rope positive, negative, or zero? Explain. (b) Calculate the work done by the rope on the skier.

Short answer

(a) Positive work. (b) 7800 J of work is done by the rope.

Step-by-step solution

Understand Work Concepts

Work is done when a force acts on an object to move it in the direction of the force. The work done by the force is calculated using the formula: \[ W = F \cdot d \cdot \cos(\theta) \]where \( W \) is the work done, \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors. In this problem, note that the force by the rope and the displacement of the skier are in the same direction.

Determine Direction of Work

Since the force exerted by the rope and the displacement are parallel (same direction), the angle \( \theta = 0^\circ \). The cosine of 0 degrees is 1. Thus, the work done on the skier by the rope is positive because the skier is moving in the same direction as the force applied by the rope. Positive work increases the energy of the skier.

Calculate Work Done

Using the work formula:\[ W = F \cdot d \cdot \cos(\theta) \]we substitute the given values. The tension in the rope \( F = 120 \; \text{N} \), displacement \( d = 65 \; \text{m} \), and \( \cos(0) = 1 \). Thus, the calculation becomes:\[ W = 120 \; \text{N} \times 65 \; \text{m} \times 1 = 7800 \; \text{J} \].The work done by the rope on the skier is \( 7800 \; \text{J} \).

Key concepts

Constant Velocity

When an object moves with constant velocity, it means that it maintains a steady speed and direction over time.
This implies there is no acceleration happening; the forces acting on the object are balanced. In our scenario, the water skier is being pulled by a boat with a constant velocity.
This constant velocity indicates that the tension in the tow rope exactly counteracts any opposing forces, like water resistance or friction.

• Constant velocity means the net force is zero because initial velocity = final velocity.
• No acceleration is present since acceleration is a change in velocity.

This balance allows us to focus solely on the work done, as the skier maintains a steady pace until they fall.

Force and Displacement

In physics, force and displacement play vital roles in the concept of work.
Work is defined as the energy transferred when a force acts upon an object to move it over a distance.
The relationship between force and displacement can be calculated using the formula for work: \[ W = F \cdot d \cdot \cos(\theta) \]where:

• \( W \) is the work done in joules.
• \( F \) is the force applied in newtons.
• \( d \) is the displacement in meters.
• \( \theta \) is the angle between the force and the direction of displacement.

In this exercise, the skier moves in the direction of the rope's tension, so \( \theta = 0 \), making \( \cos(0) = 1 \).
This simplifies the calculation, emphasizing the understanding that work is being effectively done because the force used fully aligns with the skier's movement.

Tension in Rope

Tension is the force exerted along a medium, like a rope or cable, due to the forces acting at each end.
In the case of the water skier, the tension in the rope is constant as the boat pulls them.
This tension provides the necessary force to overcome water resistance and move the skier.

• The given tension here is \( 120 \; \text{N} \).
• It is critical in maintaining the skier's constant velocity.

As the rope remains taut, it ensures a direct and consistent application of force.
When analyzing problems involving tension, it is important to assume that the rope does not stretch and that the force is transmitted uniformly along its length.
This assumption helps us conclude that the skier keeps moving directly behind the boat at a consistent speed until other forces disrupt this balance, like when the skier falls.