Problem 17 - Applied Physics

Question

An underwater camera weighing \(1250 \mathrm{~N}\) in air is submerged and supported by a tether line. If the volume of the camera is \(8.30 \times 10^{-2} \mathrm{~m}^{3}\), what is the tension in the line?

Step-by-Step Solution

Answer

The tension in the line is approximately 435.77 N.

Step 1: Understand the Problem

The camera's weight in air is given as \(1250 \mathrm{~N}\). Once submerged, the camera experiences a buoyant force due to the displacement of water. We need to find the tension in the tether line which will be the difference between the camera's weight and the buoyant force acting on it.

Step 2: Calculate the Buoyant Force

The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force equals the weight of the displaced fluid. Here, the displaced fluid is water. The buoyant force formula is: \[ F_b = \rho \times V \times g \] where \( \rho \approx 1000 \mathrm{~kg/m^3} \) is the density of water, \( V = 8.30 \times 10^{-2} \mathrm{~m}^{3} \) is the volume of the camera, and \( g \approx 9.81 \mathrm{~m/s^2} \) is the acceleration due to gravity.

Step 3: Calculate Tension in the Tether Line

The tension in the line equals the weight of the camera minus the buoyant force: \[ T = W - F_b \] Calculate \( F_b \) using the values from Step 2 and subtract it from the weight \( W = 1250 \mathrm{~N} \). Plugging in the numbers: \[ F_b = 1000 \times 8.30 \times 10^{-2} \times 9.81 = 814.23 \mathrm{~N} \] Then: \[ T = 1250 - 814.23 = 435.77 \mathrm{~N} \]

Key Concepts

Buoyant Force

When an object is submerged in a fluid, it feels a force pushing it upwards. This is called the buoyant force. The principle that explains this phenomenon is known as Archimedes' principle. It states that the buoyant force on an object submerged in a fluid equals the weight of the fluid it displaces. Knowing this concept helps us calculate how objects behave in water, like our underwater camera in the exercise.

Here's how we calculate the buoyant force:

• You need the density of the fluid, which for water is approximately \(1000\, \text{kg/m}^3\).
• Next, use the volume of the submerged object. In our case, it's \(8.30 \times 10^{-2} \text{ m}^3\) for the camera.
• Finally, multiply these values by the gravitational acceleration (\( g \approx 9.81 \text{ m/s}^2\)).

This product gives the buoyant force. Understanding buoyant force is crucial to solving problems involving objects in fluid.

Tether Line Tension

The tether line tension is what keeps our submerged camera from being completely buoyant. This tension results from the balance between the camera's weight and the upward buoyant force.

When calculating the tension:

• Start with the weight of the object in air, here it's \(1250\, \text{N}\).
• Next, subtract the buoyant force from this weight. The buoyant force calculated was \(814.23\, \text{N}\).
• The difference gives you the tension in the tether line. This makes sense because the camera under water is effectively lighter by the amount of the buoyant force.

This tension calculation is crucial, particularly for items like cameras, as a wrong tension can lead to the camera drifting away or getting damaged.

Volume Displacement

Volume displacement is a core concept when talking about buoyancy. When an object like our camera submerges in water, it pushes aside or displaces a quantity of water equal to its own volume.

This displaced volume is key because:

• It determines the amount of fluid's weight that is pushing up against the object.
• It's a central part of finding the buoyant force since the buoyant force equals the weight of this displaced fluid.

In our exercise, the camera displaced \(8.30 \times 10^{-2}\, \text{m}^3\) of water. This simple idea explains why objects float or sink, revealing the importance of understanding displaced volume in exercises like these.