Question

Think \& Calculate 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. (b) 7800 J.

Step-by-step solution

Identify the Type of Work Done

The tension in the tow rope pulls the skier in the same direction as the motion. Since the direction of the force (tension in the rope) and the direction of movement are the same, the work done is positive. When force and movement are in the same direction, work is considered positive.

Apply the Work Formula

To calculate the work done, use the formula: \[ Work = Force \times Distance \]Here, the force (tension in the rope) is 120 N, and the distance the skier is pulled is 65 m. This simplifies to:\[ Work = 120 \, \text{N} \times 65 \, \text{m} \]

Calculate the Work Done

Perform the multiplication to find the work done:\[ Work = 120 \cdot 65 = 7800 \, \text{J} \]Thus, the work done by the rope on the skier is 7800 Joules.

Key concepts

Tension Force

Tension force is a type of force that is transmitted through a string, rope, cable, or any other flexible connector when it is pulled tight by forces acting from opposite ends. This force pulls equally on all connected objects in the opposite direction. In the given physics problem, the tow rope applies a tension force of \(120 \, \text{N}\) to the skier. This means that the rope is pulling the skier with this force while remaining taut. Tension is crucial in situations where connecting elements, such as ropes or cables, are used to transmit force from one object to another. \(
\)

• It acts along the length of the rope or cable.
• It is always directed away from the object it acts upon.

Work Formula

The work formula is a fundamental concept in physics, used to calculate the amount of energy transferred by a force moving an object over a distance. The basic formula for work is given by:\[ \text{Work} = \text{Force} \times \text{Distance} \times \cos(\theta) \]where \(\theta\) is the angle between the force direction and displacement. In this scenario, however, the force and displacement are in a straight line, so \(\cos(\theta) = 1\). Thus, the formula simplifies to \[ \text{Work} = \text{Force} \times \text{Distance} \]\(
\)

• Force is measured in newtons (\text{N})
• Distance is how far the object moves while the force acts, measured in meters (m)

Using this formula, the work done by the rope on the skier is calculated as \(120 \, \text{N} \times 65 \, \text{m} = 7800 \, \text{J}\) (Joules), indicating energy transferred to the skier.

Positive Work

Positive work occurs when the force applied to an object is in the same direction as the object's movement. This means the object gains energy, increasing its kinetic energy, due to the work done. In our problem, since the tension force in the rope and the skier's movement are in the same direction, the work performed by the rope is considered positive. \(
\)The concept of positive work is important because it indicates an increase in the object's energy. When you think about work in terms of energy, it becomes easier to understand why the tow rope is doing positive work on the skier. The skier is being pulled forward, and their kinetic energy increases due to this energy transfer.

Physics Problem-Solving

Solving physics problems often involves understanding the forces at play and how they relate to movement and energy. This requires a step-by-step approach to clearly identify each element and apply the appropriate formulas. \(
\)To effectively solve problems like calculating work:

• Identify the forces involved and their direction.
• Determine the type of work (positive or negative).
• Apply the correct formulas with the given values.
• Perform calculations accurately to find the answer.

By breaking down the problem into these manageable steps, you can methodically work through it without becoming overwhelmed, ensuring accuracy and a deeper understanding of the concepts involved. The process of applying formulas and interpreting the results builds a strong foundation in physics problem-solving.