07 Work

 

Learning Outcomes

• Solve problems involving work and energy in marine contexts; and
• Solve problems involving conservation of mechanical energy

Work in Physics and Marine Biology

Introduction to Work in Physics

In physics, work is done when a force is applied to an object, and the object moves in the direction of that force. The amount of work done depends on three factors: the force applied, the distance the object moves, and the direction of the force relative to the direction of motion. The formula for calculating work is:

W = F ⋅ d ⋅ cos(θ)

Where:

• W is the work done (in joules, J),
• F is the force applied (in newtons, N),
• d is the distance moved (in meters, m),
• θ is the angle between the force and the direction of motion.

If the force is applied in the same direction as the motion (θ = 0°), the equation simplifies to W=F⋅d. If the force is perpendicular to the direction of motion (θ = 90°), no work is done since cos(90°)=0.

Work and Energy Relationship

Work is a way of transferring energy from one object to another. When work is done on an object, its energy changes. For example, lifting a marine organism or equipment from the seabed to the boat involves doing work against gravity, which increases the object’s potential energy. Similarly, work can increase the kinetic energy of an object when a force accelerates it.

Sample Problem 1: Lifting a Fish Trap

A marine biologist working in Cantaan, Camiguin, lifts a fish trap weighing 50 N from the seabed to the deck of a boat, which is 2 meters above the seabed. How much work does the biologist do in lifting the trap?

Given:
Force applied (F) = 50 N
Distance moved (d) = 2 m
Angle (θ) = 0° (the force and movement are in the same direction, vertically)

Find: W

Solution:

Using the formula:

W = F⋅d⋅cos(θ) = 50 N × 2 m × cos(0^∘) = 50 N × 2 m × 1 = 100 J

The marine biologist does 100 joules of work to lift the fish trap.

Real-Life Marine Biology Research Application: Work in Marine Conservation

In marine conservation, work is frequently done in the form of moving heavy equipment and organisms during research activities. For example, researchers in Cantaan, Camiguin, often move giant clams for conservation purposes. The clams need to be transferred from one location to another to ensure their safety or to monitor their growth in different marine environments.

Imagine moving a giant clam weighing 500 N from the seabed to a boat 3 meters above. The work done can be calculated to ensure the appropriate force is used during transportation. This ensures that the researchers understand how much energy is required, helping them plan their equipment needs efficiently and preventing strain or accidents.

Sample Problem 2: Moving a Giant Clam

A team of marine biologists needs to lift a giant clam with a weight of 500 N from the seabed to a boat 3 meters above. Calculate the work done in lifting the clam.

Given:
Force applied (F) = 500 N
Distance moved (d) = 3 m
Angle (θ) = 0°

Find: W

Solution:

W = F⋅d⋅cos(θ) = 500 N × 3 m × 1 = 1,500 J

The team does 1,500 joules of work to lift the giant clam.

Practice Problems

1. A marine biologist pulls a net horizontally along the seashore with a force of 100 N. The net moves 5 meters in the direction of the force. How much work is done by the biologist?

2. A diver applies a force of 200 N to move a piece of equipment underwater. The equipment moves 1.5 meters at an angle of 30° to the force. How much work is done on the equipment?

3. A scientist hauls a sample of marine sediment weighing 250 N vertically from the seabed to a boat 4 meters above. How much work is done?

4. A fishing boat drags a net horizontally through the water with a force of 400 N. The net moves 10 meters. How much work is done on the net?

Work in Boats and Marine Equipment

In marine environments, work is commonly done by boats when they move through water. The engines of boats apply force to push water backward, and this force propels the boat forward. The amount of work done by the boat's engine depends on the distance it travels and the resistance it faces from the water (drag force). Understanding this relationship helps marine biologists and engineers design more efficient boats, improving fuel efficiency and reducing environmental impact.

Sample Problem 3: Work Done by a Boat

A small fishing boat applies a force of 800 N to move forward through the water. If the boat moves 50 meters, calculate the work done by the boat's engine.

Given:
Force applied (F) = 800 N
Distance moved (d) = 50 m
Angle (θ) = 0° (the force is in the direction of motion)

Find: W

Solution:

W = F⋅d⋅cos(θ) = 800 N × 50 m × 1 = 40,000 J

The boat’s engine does 40,000 joules of work to move the boat 50 meters through the water.

Advanced Example: Work Against Drag Forces in Tidal Energy Research

Marine researchers are increasingly interested in harnessing tidal energy to power coastal communities. Tidal turbines are installed underwater to convert the kinetic energy of moving water into electrical energy. To understand how much work is done by the water on the turbines, engineers calculate the force exerted by the water flow and the distance over which the turbines interact with the current.

Sample Problem 4: Tidal Turbine Work

A tidal turbine experiences a water drag force of 2,000 N as water flows through it. If the water flows a distance of 20 meters past the turbine, how much work does the water do on the turbine?

Given:
Force applied by water (F) = 2,000 N
Distance moved by water (d) = 20 m
Angle (θ) = 0° (force and water movement are aligned)

Find: W

Solution:

W = F⋅d⋅cos(θ) = 2,000 N × 20 m × 1 = 40,000J

The water does 40,000 joules of work on the turbine.

Sample Problem 5: Work with an Angle

A marine biologist is pulling a water sample collector on the beach with a force of 200 N. The rope is inclined at an angle of 30° to the horizontal, and the sample collector is dragged 50 meters across the sand. Calculate the work done by the biologist in moving the collector.

Given:

F = 200 N
d = 50 m
θ = 30^∘
cos30^∘ ​​≈ 0.866

Find: W

Solution:

W = F d cosθ = 200 N × 50 m × 0.866 = 8,660J

The work done by the biologist is 8,660 joules.

Sample Problem 6: Work and Kinematic Equations

A sea turtle, with a mass of 250 kg, is swimming through a current. It accelerates from rest to a speed of 3 m/s over a distance of 20 meters. Calculate the net work done on the turtle.

Given:

v_f = 3 m/s
v_i = 0 m/s
d = 20 m
m = 250 kg

Find: W

Solution:

1. Use the kinematic equation to find acceleration first: v²=v^_o2+2ad

Solving for a:

3² = 0 + 2a(20)
a = 9/40 = 0.225 m/s²

2. Now, use Newton's second law to find the net force:

F = ma = 250 × 0.225 = 56.25 N

3. Finally, use the work formula to calculate the net work:

W = Fd = 56.25 × 20 = 1,125 J

The net work done on the turtle is 1,125 joules.

Sample Problem 7: Work and Newton’s Laws of Motion

A 5,000 kg boat is initially at rest. A motor applies a force of 2,000 N to the boat, causing it to move along a straight path for 10 seconds. Calculate the work done on the boat by the motor.

Given:

m = 5,000 kg
v_o = 0
F = 2000 N
t = 10 s

Find: W

Solution:

1. Use Newton’s second law to find the boat’s acceleration:

F = ma

Solving for a:

2,000 = 5,000 × a ⟹ a = 0.4 m/s²

2. Use the kinematic equation to find the displacement during the 10 seconds:

d = v_o t + (1/2)at²

d = (1/2) ​× 0.4 × (10)2 = 20m

3. Now, calculate the work done:

W = Fd = 2,000 × 20 = 40,000 J

The work done on the boat by the motor is 40,000 joules.

Practice Problems

5. A marine engineer uses a rope to pull a 400 kg buoy with a force of 500 N at an angle of 45° to the horizontal. The buoy is dragged for 30 meters. Calculate the work done by the engineer.

6. A 300 kg sea lion starts from rest and accelerates to 4 m/s over a distance of 15 meters. Calculate the net work done on the sea lion.

7. A diver’s boat with a mass of 3,000 kg is pushed by a constant force of 1,500 N over 12 seconds. Calculate the work done on the boat after the diver pushes it for the given time.

These problems will help students apply their understanding of work in marine environments and prepare them for real-world applications in physics and marine biology research.

Work is a fundamental concept that plays an essential role in marine biology and environmental research. From lifting equipment and moving marine organisms to understanding tidal energy, calculating the work done helps marine scientists plan their activities efficiently. In your upcoming laboratory exercise, you will practice solving work-related problems in a variety of real-world marine contexts to strengthen your problem-solving skills.