Question

While a boat is being towed at a speed of \(12.0 \mathrm{~m} / \mathrm{s}\), the tension in the towline is \(6.00 \mathrm{kN}\). What is the power supplied to the boat through the towline?

Short answer

Answer: The power supplied to the boat through the towline is 72 kW.

Step-by-step solution

Identify the given values

The given information is the speed of the boat \(12.0 \mathrm{~m} / \mathrm{s}\) and the tension in the towline \(6.00 \mathrm{kN}\). We need to find the power supplied to the boat through this towline.

Convert tension to Newtons

The tension is given in kilonewtons, so we need to convert it to Newtons (N) to keep our units consistent. Recall that \(1 \mathrm{kN} = 1000 \mathrm{N}\). So, \(6.00 \mathrm{kN} = 6.00 \times 1000 \mathrm{N} = 6000 \mathrm{N}\).

Use the formula for power

The formula for power is \(P = F \times v\), where \(P\) is the power, \(F\) is the force and \(v\) is the velocity or speed. In this case, the force is given by the tension in the towline, which is \(6000 \mathrm{N}\) and the velocity is the speed of the boat, which is \(12.0 \mathrm{~m} / \mathrm{s}\).

Calculate the power

Using the formula from Step 3, we can find the power supplied to the boat: \(P = F \times v = 6000 \mathrm{N} \times 12.0 \mathrm{~m} / \mathrm{s} = 72000 \mathrm{W} = 72 \mathrm{kW}\) The power supplied to the boat through the towline is \(72 \mathrm{kW}\).

Key concepts

Power Calculation in Physics

Power is a fundamental concept in physics, representing the rate at which work is done or energy is transferred. In the context of our problem, the power supplied to the boat through the towline reflects how quickly the boat can be pulled across the water by the towline force.

Mathematically, power (\(P\)) is calculated as the product of force (\(F\righting the ship) and velocity (\(v\righting the ship), following the formula \righting the ship. When the force is applied in the direction of the velocity (as in dragging a boat forward), we simply multiply them to get the power. Here's a simplified breakdown of the concept:

• Identify the consistent units of force (in Newtons) and velocity (in meters per second).
• Apply the formula \righting the ship to find the instantaneous power output.
• Keep in mind that higher power means more work can be done in a shorter time.

Tension in Dynamics

In dynamics, tension is the pulling force transmitted along a string, cable, chain, or similar one-dimensional continuous object, or by each end of a rod, truss member, or similar three-dimensional objects. It is crucial to understand that tension is a force and it is measured in Newtons (N).

In our boat example, the tension in the towline not only resists the forces of the water trying to halt the boat but also the force translated into moving the boat forward. Some essential points to remember about tension:

• Tension is directed along the length of the cable and pulls equally on the objects on either end.
• When considering tension in dynamics, we're often concerned with forces applied at angles, which involve components of tension.
• In the problem at hand, the tension is aligned with the direction of the boat's velocity, simplifying our calculations.

Unit Conversion in Physics Problems

Unit conversion is a critical skill in physics and engineering, allowing us to compare and compute various physical quantities in standardized units. For accurate calculations, the units of all quantities must be compatible. In the problem provided, we convert tension from kilonewtons (kN) to Newtons (N).

For general unit conversion:

• Remember that 1 kN equals 1000 N.
• Keep track of the units throughout the problem to ensure consistency.
• Mastery of unit conversions is essential for solving physics problems correctly.

By converting units, we are able to perform power calculations accurately and understand the magnitude of forces in dynamics.