Problem 3 - College Physics

Question

\(\cdot\) A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of \(15.0^{\circ}\) with the forward direction of motion. If the tension in the rope is \(180 \mathrm{N},\) how much work does the rope do on the skier during a forward displacement of 300.0 \(\mathrm{m} ?\)

Step-by-Step Solution

Answer

The work done is approximately 52158.6 J.

Step 1: Define Work Formula

The work done by a force is given by the formula \( W = F \cdot d \cdot \cos(\theta) \), where \( W \) is the work, \( F \) is the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and the direction of displacement.

Step 2: Identify Known Values

From the problem, the tension in the rope \( F = 180 \, \text{N} \), the forward displacement \( d = 300.0 \, \text{m} \), and the angle \( \theta = 15.0^{\circ} \).

Step 3: Calculate Work Using Given Values

Substitute the known values into the work formula: \[ W = 180 \, \text{N} \times 300.0 \, \text{m} \times \cos(15.0^{\circ}) \].

Step 4: Compute Cosine of the Angle

Calculate \( \cos(15.0^{\circ}) \), which is approximately \( 0.9659 \).

Step 5: Final Calculation of Work

Multiply the values together: \[ W = 180 \, \text{N} \times 300.0 \, \text{m} \times 0.9659 \]. This results in \( W \approx 52158.6 \, \text{J} \).

Key Concepts

Work Formula

Understanding the work formula helps us determine how much energy is transferred by a force acting over a distance. The work done by a force is given by the formula:\[ W = F \cdot d \cdot \cos(\theta) \]Here,

• \( W \) represents the work done in Joules (J).
• \( F \) is the magnitude of the force applied in Newtons (N).
• \( d \) is the displacement, or the distance moved by the object, measured in meters (m).
• \( \theta \) is the angle between the force vector and the direction of displacement, measured in degrees.

By decomposing the vector into its components, the work formula accounts not only for the magnitude of the force and distance but also how aligned the force is with the direction of displacement. This insight is key to understanding the effectiveness of the force in doing work.

Force and Displacement

To better grasp how force relates to displacement, consider force as a push or pull applied to an object. Displacement, on the other hand, is how far the object moves and in what direction. In our example with the water skier, the rope applies a force of 180 N which pulls the skier 300.0 m in the forward direction.
However, whenever you consider work being done, it’s crucial to also think about the direction in which the force is applied. If the force and displacement are in the same direction, the full force contributes to doing the work. If the force is only partially aligned with the displacement, such as forming an angle with the path, only a portion of the force effectively contributes to moving the object forward. This nuance is captured in our work formula by the cosine of the angle between force and displacement.

Angle of Force Application

The angle of force application is a critical factor in determining the actual work done by a force. In the context of work, \( \theta \) is the angle between the force vector and the direction of the displacement. The angle affects how much of the force actually contributes to the movement of the object.
A

Role of Cosine in the Work Formula

Cosine is used in the work formula to account for how much of the force is effectively applied in the direction of the displacement. When the angle is zero (meaning the force is perfectly aligned with the displacement), \( \cos(\theta) \) is 1, so all the force contributes to work done. As the angle increases, \( \cos(\theta) \) decreases, and thus, less of the force contributes to the actual work.
If we return to our example, the tow rope makes a \( 15.0^{\circ} \) angle with the direction of motion. Therefore, the calculation involves multiplying the force by \( \cos(15.0^{\circ}) \), which is approximately 0.9659. This means that around 96.59% of the force is effective in doing work forward. Understanding this relationship between angle and work helps us calculate the energy efficacy in practical scenarios.