Problem 84 - University Physics with Modern Physics

Question

A deep-sea diver is suspended beneath the surface of Loch Ness by a \(100-\) m-long cable that is attached to a boat on the surface (Fig. P15.84). The diver and his suit have a total mass of 120 \(\mathrm{kg}\) and a volume of 0.0880 \(\mathrm{m}^{3} .\) The cable has a diameter of 2.00 \(\mathrm{cm}\) and a linear mass density of \(\mu=\) 1.10 \(\mathrm{kg} / \mathrm{m} .\) The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat. (a) What is the tension in the cable at its lower end, where it is attached to the diver? Do not forget to include the buoyant force that the water (density 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) ) exerts on him. (b)Calculate the tension in the cable a distance \(x\) above the diver. The buoyant force on the cable must be included in your calculation. (c) The speed of transverse waves on the cable is given by \(v=\sqrt{F / \mu}\) (Eq. 15.13). The speed therefore varies along the cable, since the tension is not constant. (This expression neglects the damping force that the water exerts on the moving cable.. Integrate to find the time required for the first signal to reach the surface.

Step-by-Step Solution

Answer

The tension at the diver's end is greater than just the weight due to the buoyancy, and the signal takes time determined by varying tension along the cable.

Step 1: Calculate Buoyant Force on Diver

The buoyant force is equal to the weight of water displaced by the diver's volume. Use the formula for buoyant force: \[ F_b = \rho_{water} \times V_{diver} \times g \] where \( \rho_{water} = 1000 \, \text{kg/m}^3 \), \( V_{diver} = 0.0880 \, \text{m}^3 \), and \( g = 9.8 \, \text{m/s}^2 \). Calculate \( F_b \).

Step 2: Calculate Weight of Diver

The weight of the diver is the product of his mass and gravitational acceleration: \[ W_{diver} = m_{diver} \times g \] where \( m_{diver} = 120 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). Calculate \( W_{diver} \).

Step 3: Calculate Tension at Lower End of Cable

The tension in the cable at the lower end is the difference between the diver's weight and the buoyant force:\[ T = W_{diver} - F_b \] Substitute the values of \( W_{diver} \) and \( F_b \) from Steps 1 and 2 to calculate the tension \( T \).

Step 4: Calculate Tension at a Distance x Above the Diver

The tension \( T_x \) a distance \( x \) above the diver includes the weight of the segment of cable below this point and the adjusted buoyant force:\[ T_x = T + (\text{weight of cable segment} - \text{buoyant force on cable segment}) \] Calculate the mass of the cable segment as \( m_{cable} = \mu \times x \) and find its weight. Then find the buoyant force on the cable segment, \( F_{b,cable} = \rho_{water} \times \pi \times (\frac{d}{2})^2 \times x \times g \). Substitute these into the equation for \( T_x \).

Step 5: Integrate to Find Time for Signal to Reach Surface

The wave speed \( v(x) \) at point \( x \) is given by the expression:\[ v(x) = \sqrt{\frac{T(x)}{\mu}} \] To find the time \( t \) for the wave to travel from the diver to the surface:\[ t = \int_0^{100} \frac{1}{v(x)} \, dx \] Use the expression for \( T(x) \) from Step 4 in \( v(x) \) and perform the integration to calculate the time \( t \).

Step 6: Solve the Integral

Substitute the expression for \( T(x) \) from Step 4 into the wave velocity equation:\[ v(x) = \sqrt{\frac{T + (\mu \times x \times g - \rho_{water} \times \pi \times (\frac{d}{2})^2 \times x \times g)}{\mu}} \] Then solve the integral:\[ t = \int_0^{100} \frac{1}{\sqrt{\frac{T + (\mu \times x \times g - \rho_{water} \times \pi \times (\frac{d}{2})^2 \times x \times g)}{\mu}}} \, dx \] Perform the integration to find the value of \( t \). Utilize numerical methods if required for simplification.

Key Concepts

Tension in Cable

Tension in a cable is a critical concept when studying wave propagation and other physical phenomena involving cables or strings. In the context of our exercise, the tension is crucial in determining how forces are transmitted along a cable connecting objects. For example, the diver hanging beneath the water's surface affects the tension, which is influenced by both the weight of the diver and the buoyant force exerted by the water.

Key points to keep in mind:

• The tension at the lower end of the cable is the difference between the diver's weight and the buoyant force.
• Tension varies along the cable, from the diver to the boat on the surface, due to the non-uniform distribution of forces like weight and buoyancy.
• This variance is important because it directly affects the speed at which transverse waves travel along the cable.

Understanding these factors can help in calculating the wave propagation time and effectively communicating signals using wave mechanics.

Buoyant Force

The buoyant force is a fundamental concept in fluid mechanics, describing the upward force exerted by a fluid on an object submerged in it. This force is crucial in our exercise as it impacts the net force acting on the diver and the cable.

Some essential aspects of buoyant force:

• It's calculated as the weight of the fluid displaced by the object, using the formula: \( F_b = \rho_{water} \times V_{diver} \times g \).
• The magnitude of this force depends on the density of the fluid (in this case, water) and the volume of the object submerged.
• Buoyant force reduces the effective weight of the diver in water, thus affecting the tension in the cable.

This knowledge is critical in predicting and adjusting for the actual forces experienced by the diver and how they influence the cable dynamics.

Transverse Waves

Transverse waves are types of mechanical waves where particle displacement is perpendicular to the direction of wave propagation. In our scenario, these waves travel along the cable when the diver sends a signal. Their behavior is influenced by the tension in the cable and the medium it moves through.

Important characteristics of transverse waves:

• The wave speed depends on the tension in the cable and its linear mass density.
• High tension means higher wave speeds, while low tension reduces speed.
• The formula used to calculate wave speed is \( v = \sqrt{\frac{F}{\mu}} \).

Understanding how tension affects wave propagation is essential for calculating the time it takes signals to reach the surface, especially when the tension is not uniform along the cable's length.

Integration in Physics

Integration is a powerful mathematical tool used to find areas under curves, amongst other applications. In physics, it becomes invaluable in calculating quantities like time in complex systems where variables change over a certain range.

In this exercise, integration helps to determine:

• The total time it takes for signals to travel from the diver to the surface, given the varying wave speed along the cable.
• Integration accounts for changes in tension as a function of position, providing an accurate time calculation.
• Specifically, using the integral \[ t = \int_0^{100} \frac{1}{v(x)} \, dx \], considers the non-linear tension distribution.

Mastering integration allows one to seamlessly handle diverse scenarios in physics, where conditions are not constant, thus leading to more precise solutions.