Question

A tourist of mass \(60.0 \mathrm{~kg}\) notices a chest with a short chain attached to it at the bottom of the ocean. Imagining the riches it could contain, he decides to dive for the chest. He inhales fully, thus setting his average body density to \(945 \mathrm{~kg} / \mathrm{m}^{3}\), jumps into the ocean (with saltwater density = \(1020 \mathrm{~kg} / \mathrm{m}^{3}\) ), grabs the chain, and tries to pull the chest to the surface. Unfortunately, the chest is too heavy and will not move. Assume that the man does not touch the bottom. a) Draw the man's free-body diagram, and determine the tension on the chain. b) What mass (in kg) has a weight that is equivalent to the tension force in part (a)? c) After realizing he cannot free the chest, the tourist releases the chain. What is his upward acceleration (assuming that he simply allows the buoyant force to lift him up to the surface)?

Short answer

(b) What is the mass of an object that has the same weight as the tension? (c) If the tourist releases the chain, what is his upward acceleration? Answer: (a) The tension in the chain is 49.1 N. (b) The mass of an object with the same weight as the tension is 5.01 kg. (c) The upward acceleration of the tourist when he releases the chain is 10.63 m/s².

Step-by-step solution

Calculate the volume of the tourist

First, we will calculate the volume of the tourist. We can use the formula: $$ \text{Volume} = \frac{\text{Mass}}{\text{Density}} $$ $$ V = \frac{60.0 \thinspace kg}{945 \thinspace kg/m^3} = 0.06349 \thinspace m^3 $$

Calculate the buoyant force

Next, we need to calculate the buoyant force on the tourist. According to Archimedes' principle, the buoyant force is equal to the weight of the liquid displaced by the body. The formula for the buoyant force is: $$ F_b = \rho_{\text{fluid}} \times V \times g $$ Where \(\rho_{\text{fluid}}\) is the density of saltwater, \(V\) is the volume of the tourist, and \(g\) is the acceleration due to gravity (\(9.81 \thinspace m/s^2\)). $$ F_b = 1020 \thinspace kg/m^3 \times 0.06349 \thinspace m^3 \times 9.81 \thinspace m/s^2 = 637.7 \thinspace N $$

Calculate the weight of the tourist

Now, we will calculate the weight of the tourist using the formula: $$ W = m \times g $$ Where \(m\) is the mass of the tourist and \(g\) is the acceleration due to gravity. $$ W = 60.0 \thinspace kg \times 9.81 \thinspace m/s^2 = 588.6 \thinspace N $$

Determine the tension on the chain

Finally, we will determine the tension on the chain by finding the difference between the buoyant force and the weight of the tourist: $$ T = F_b - W $$ $$ T = 637.7 \thinspace N - 588.6 \thinspace N = 49.1 \thinspace N $$ The tension on the chain is 49.1 N. b) Determine the mass with equivalent weight:

Calculate the mass with equivalent weight

To find the mass with a weight equivalent to the tension force, we can use the formula: $$ m = \frac{W}{g} $$ Where \(W\) is the tension force (49.1 N) and \(g\) is the acceleration due to gravity (9.81 m/s^2). $$ m = \frac{49.1 \thinspace N}{9.81 \thinspace m/s^2} = 5.01 \thinspace kg $$ The mass with an equivalent weight to the tension force is 5.01 kg. c) Calculate the upward acceleration:

Calculate the net upward force

The net upward force on the tourist when the chain is released is equal to the buoyant force: $$ F_{\text{net}} = F_b = 637.7 \thinspace N $$

Calculate the upward acceleration

To calculate the upward acceleration, we will use Newton's second law of motion: $$ F = m \times a $$ Solving for acceleration, we have: $$ a = \frac{F}{m} $$ We will use the net upward force (637.7 N) and the mass of the tourist (60 kg) in this formula. $$ a = \frac{637.7 \thinspace N}{60 \thinspace kg} = 10.63 \thinspace m/s^2 $$ The upward acceleration of the tourist when he releases the chain is 10.63 m/s².

Key concepts

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object submerged in it. This concept is derived from Archimedes' Principle, which states that the buoyant force on an object is equal to the weight of the fluid the object displaces.

In the exercise, a tourist with a mass of 60 kg jumps into the ocean, and his buoyant force is calculated by the formula:\[F_b = \rho_{\text{fluid}} \times V \times g\]Here,

• \(\rho_{\text{fluid}}\) is the density of saltwater (1020 kg/m³),
• \(V\) is the volume of the tourist, which is calculated as mass (60 kg) divided by his density (945 kg/m³),
• \(g\) is the acceleration due to gravity (9.81 m/s²).

After plugging these values into the formula, the buoyant force exerted on the tourist is found to be 637.7 N. This force provides the upward push necessary for flotation and plays a critical role in determining whether the tourist floats, sinks, or remains neutrally buoyant while submerged.

Free-Body Diagram

A free-body diagram is a crucial tool used in physics to show all the forces acting on an object. These diagrams help visualize the situation and calculate unknown forces.

For the tourist in the ocean, the key forces depicted on the free-body diagram are:

• The buoyant force \(F_b\) of 637.7 N acting upwards, exerted by the saltwater.
• The weight \(W\) of the tourist acting downwards, calculated as \(60 \thinspace kg \times 9.81 \thinspace m/s^2 = 588.6 \thinspace N\).
• The tension force \(T\) from holding onto the chest chain, also acting downwards.

The free-body diagram helps us understand how these forces interact and provides a visual way to tackle force balance equations, such as \( T = F_b - W \), to determine unknowns like the tension in the chain.

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle that states an object's acceleration is dependent on the net force acting on it and inversely proportional to its mass. Mathematically, this is expressed as \(F = m \times a\).

In the given exercise, after releasing the chain, the tourist experiences a net upward force equal to the buoyant force \(F_b\), since other downward forces are taken into account. We calculate his upward acceleration by the formula:\[a = \frac{F_{\text{net}}}{m}\]Applying this,

• \(F_{\text{net}}\) is the buoyant force of 637.7 N.
• \(m\) is the mass of the tourist (60 kg).

The calculated upward acceleration is 10.63 m/s². This indicates how quickly the tourist will rise to the surface, demonstrating the practical application of Newton's second law in calculating dynamic motion scenarios.

Density Calculation

Density is a measure of how much mass is contained in a given volume, expressed as mass per unit volume \(\rho = \frac{m}{V}\). A critical aspect of the exercise is the tourist's body density, which affects his buoyancy.

The body density in the problem is given as 945 kg/m³ after the tourist inhales fully, which affects the calculation of his volume:\[V = \frac{m}{\rho} = \frac{60 \thinspace kg}{945 \thinspace kg/m^3}\]Using this calculation, the tourist's volume is found to be 0.06349 m³.
Knowing the volume allows us to apply Archimedes' Principle to determine the buoyant force acting on the tourist. The difference in density between the tourist and saltwater (1020 kg/m³) critically influences the force balance, as higher density fluid promotes flotation, which leads to the upward buoyant force observed.