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What is the Application of Vectors in Work Done by a Force?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The application of vectors in work done by a force helps us calculate how much energy is transferred when a force moves an object. It uses the 'dot product' of the force vector and the displacement vector to find the work done, which is a scalar quantity (just a number, no direction).
Simple Example
Quick Example
Imagine pushing a heavy dabba (container) across the floor. If you push it straight ahead, all your effort goes into moving it. But if you push it at an angle, only part of your push actually helps it move forward. Vectors help us figure out exactly how much of your push (force) is effective in moving the dabba (displacement).
Worked Example
Step-by-Step
Let's say a force of 10 Newtons (N) pushes a toy car 5 meters (m) forward.
Step 1: Identify the force vector (F) and displacement vector (d).
F = 10 N (acting horizontally)
d = 5 m (also horizontally)
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Step 2: Since the force and displacement are in the same direction, the angle between them is 0 degrees.
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Step 3: Recall the formula for work done using vectors: W = F . d = |F| |d| cos(theta), where theta is the angle between F and d.
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Step 4: Substitute the values into the formula.
W = (10 N) * (5 m) * cos(0 degrees)
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Step 5: Calculate cos(0 degrees), which is 1.
W = 10 * 5 * 1
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Step 6: Calculate the work done.
W = 50 Joules (J)
Answer: The work done is 50 Joules.
Why It Matters
Understanding vector applications in work done is crucial for designing everything from efficient engines in EVs to calculating rocket trajectories for ISRO. Engineers use this to ensure machines do their job effectively, while physicists use it to study energy transfer in complex systems. It's a foundational concept for careers in engineering and space science.
Common Mistakes
MISTAKE: Assuming work is always Force x Displacement, ignoring the angle. | CORRECTION: Always use the dot product formula, W = |F| |d| cos(theta), as the angle between force and displacement is critical.
MISTAKE: Confusing work done with force or displacement. | CORRECTION: Work done is a scalar quantity (energy transferred), while force and displacement are vector quantities (having magnitude and direction).
MISTAKE: Forgetting the units for work done. | CORRECTION: Work done is measured in Joules (J), which is equivalent to Newton-meters (N.m). Always include correct units in your answer.
Practice Questions
Try It Yourself
QUESTION: A force of 20 N pushes a box 3 m in the same direction as the force. What is the work done? | ANSWER: 60 J
QUESTION: A coolie pulls a suitcase with a force of 50 N at an angle of 60 degrees to the horizontal, moving it 10 m. How much work is done by the coolie? (Hint: cos(60 degrees) = 0.5) | ANSWER: 250 J
QUESTION: A boy pushes a toy car with a force vector F = (3i + 4j) N, and the car moves with a displacement vector d = (5i - 2j) m. Calculate the work done. | ANSWER: 7 J
MCQ
Quick Quiz
When is the work done by a force zero?
When the force is parallel to the displacement
When the force is perpendicular to the displacement
When the force is in the opposite direction to the displacement
When the angle between force and displacement is 45 degrees
The Correct Answer Is:
B
Work done is zero when the force is perpendicular to the displacement because cos(90 degrees) = 0. Options A, C, and D result in non-zero work done.
Real World Connection
In the Real World
Think about a construction crane lifting heavy steel beams. The crane's engine applies a force, and the beam moves upwards (displacement). Engineers use vector calculations to ensure the crane has enough power (work done) to lift the beam safely and efficiently. This is vital for urban development projects across India.
Key Vocabulary
Key Terms
VECTOR: A quantity with both magnitude and direction | SCALAR: A quantity with only magnitude (no direction) | WORK DONE: The energy transferred when a force causes displacement | DOT PRODUCT: A way to multiply two vectors, resulting in a scalar | JOULE: The SI unit of energy and work done
What's Next
What to Learn Next
Now that you understand work done by a constant force, you can explore 'Work Done by a Variable Force' using integration. This will help you tackle more complex real-world problems where forces aren't always steady, like a rocket burning fuel.
