What is Work Done by a Force (Dot Product)? | Class 12

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What is the Work Done by a Force using Dot Product?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The work done by a force is a measure of energy transfer when a force causes an object to move over a distance. When the force and displacement are vectors, we use the dot product (also called scalar product) to find the work done, as work itself is a scalar quantity.

Simple Example

Quick Example

Imagine you are pushing a heavy cart at a railway station. If you push the cart with a force of 10 Newtons (N) and it moves 5 meters (m) in the same direction you are pushing, the work done is 10 N * 5 m = 50 Joules (J). If you push the cart at an angle, the work done will be less because only the part of your push that is in the direction of movement actually helps move the cart forward.

Worked Example

Step-by-Step

PROBLEM: A delivery person pulls a trolley with a force of 50 N at an angle of 30 degrees to the horizontal. If the trolley moves a distance of 10 m horizontally, calculate the work done.

STEP 1: Identify the given values.
Force (F) = 50 N
Displacement (d) = 10 m
Angle (theta) = 30 degrees

---STEP 2: Recall the formula for work done using the dot product.
Work (W) = F . d = |F| |d| cos(theta)

---STEP 3: Substitute the values into the formula.
W = 50 N * 10 m * cos(30 degrees)

---STEP 4: Calculate the value of cos(30 degrees).
cos(30 degrees) = 0.866 (approximately)

---STEP 5: Perform the multiplication.
W = 50 * 10 * 0.866
W = 500 * 0.866
W = 433 J

ANSWER: The work done by the delivery person is 433 Joules.

Why It Matters

Understanding work done is crucial in engineering, from designing efficient EVs to calculating rocket trajectories in Space Technology. Engineers use this concept to optimize machine performance and energy consumption, leading to innovations like faster trains and more powerful drones.

Common Mistakes

MISTAKE: Forgetting that work is a scalar quantity, not a vector. | CORRECTION: Work done (W) is always a single number (magnitude) with a unit of Joules, not a direction. The dot product of two vectors gives a scalar.

MISTAKE: Using the angle between the force and displacement incorrectly, or not using cosine. | CORRECTION: Always use the angle (theta) BETWEEN the force vector and the displacement vector. The formula is Fd cos(theta).

MISTAKE: Confusing work done with just applying a force. | CORRECTION: Work is only done if there is displacement in the direction of the force. If you push a wall, you apply force but do no work because the wall doesn't move.

Practice Questions

Try It Yourself

QUESTION: A coolie lifts a luggage bag weighing 20 kg to a height of 1.5 m. Assuming g = 10 m/s^2, what is the work done against gravity? (Hint: Force = mass x gravity) | ANSWER: Force = 20 kg * 10 m/s^2 = 200 N. Work = Force * Displacement = 200 N * 1.5 m = 300 J.

QUESTION: A child pulls a toy car with a string. The string makes an angle of 60 degrees with the ground. If the child applies a force of 15 N and the car moves 4 m, what is the work done? | ANSWER: W = Fd cos(theta) = 15 N * 4 m * cos(60 degrees) = 15 * 4 * 0.5 = 30 J.

QUESTION: A robot arm applies a force vector F = (3i + 4j) N to move an object through a displacement vector d = (2i - 1j) m. Calculate the work done by the robot arm. | ANSWER: W = F . d = (3 * 2) + (4 * -1) = 6 - 4 = 2 J.

MCQ

Quick Quiz

Which of the following is the correct unit for work done?

Newton (N)

Joule (J)

Watt (W)

Meter (m)

The Correct Answer Is:

Joule (J) is the SI unit for energy and work done. Newton is the unit for force, Watt for power, and Meter for displacement.

Real World Connection

In the Real World

In India, think about ISRO's rockets. When a rocket engine fires, it applies a massive force to propel the rocket upwards. Scientists and engineers calculate the work done by these engines to understand how much energy is needed to launch satellites into space and how far they can travel, ensuring successful missions.

Key Vocabulary

Key Terms

WORK DONE: The energy transferred when a force causes displacement. | FORCE: A push or pull on an object. | DISPLACEMENT: The change in an object's position. | DOT PRODUCT: A mathematical operation on two vectors that results in a scalar quantity. | JOULE: The SI unit of work and energy.

What's Next

What to Learn Next

Now that you understand work done, you can explore the concept of 'Power'. Power is how quickly work is done, which is super important for understanding how fast machines like electric vehicles or mobile chargers operate. Keep learning!