Day 08: Work, Energy, and Power – Mathematical Applications | Secondary Stage (Grades 9–10) Science | Apex Institute of Maths and Sciences

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Day 08: Work, Energy, and Power – Mathematical Applications | Secondary Stage (Grades 9–10) Science | Apex Institute of Maths and Sciences

Day 08: Work, Energy, and Power – Mathematical Applications

Secondary Stage (Grades 9–10) Science | Apex Institute of Maths and Sciences

Level 1: The Quest (Concept)

Welcome, Explorer! In Physics, “Work” isn’t just doing homework—it’s what happens when a Force moves an object. Energy is your capacity to do that work, and Power is how fast you can get it done! Think of Energy as your fuel tank and Power as your engine’s speed.

The Core Relationship:
If you push a block with force \(F\) over distance \(s\), Work is done: $$W = F \cdot s \cdot \cos(\theta)$$ Where \(\theta\) is the angle between force and movement.

Level 2: Power-Ups (Tools/Methods)

🚀 Master Formulas:
  • Work: \(W = Fs\) (Unit: Joule, J)
  • Kinetic Energy: \(E_k = \frac{1}{2}mv^2\)
  • Potential Energy: \(E_p = mgh\) (where \(g \approx 9.8 \, m/s^2\))
  • Power: \(P = \frac{W}{t}\) (Unit: Watt, W)
Pro-Tip: To convert Horsepower (hp) to Watts, remember: \(1 \, hp = 746 \, W\). To convert Kilowatt-hour (kWh) to Joules: \(1 \, kWh = 3.6 \times 10^6 \, J\).

Level 3: Mini-Boss Battles (Applications)

Scenario 1: The Elevator Challenge
An elevator lifts a 500kg load to a height of 20 meters in 10 seconds. Engineers use \(P = \frac{mgh}{t}\) to decide which motor strength is needed to ensure the cable doesn’t snap!
Scenario 2: The Cricket Shot
When a batsman hits a ball, the work done by the bat is converted into Kinetic Energy (\(\frac{1}{2}mv^2\)). The faster the swing (Force) and the longer the contact (Distance), the further the ball flies for a Six!

Level 4: Home Quests (Activities)

Quest 1: Stairs Power Meter
Measure the height of one step in your house. Count the steps to a floor. Record the time it takes you to run up. Calculate your personal Power output using \(P = \frac{mgh}{t}\). (Ask parents for your weight in kg!)
Quest 2: Light Bulb Audit
Check 3 different bulbs in your home. Note their Wattage (Power). Calculate how much Energy (in Joules) each consumes if left on for 1 hour (\(E = P \times 3600s\)). Discuss with parents how choosing lower watts saves money!

Final Boss: Practice Test

EASY

1. What is the SI unit of Work?

Magic Solution: Work is measured in Joules (J). Newton is for Force, Watt is for Power.
EASY

2. Kinetic energy of a body depends on:

Magic Solution: \(E_k = \frac{1}{2}mv^2\), so it depends on both \(m\) (mass) and \(v\) (velocity).
EASY

3. If the force is applied perpendicular to the direction of motion, work done is:

Magic Solution: Since \(\cos(90^\circ) = 0\), the work done is zero.
EASY

4. Power is defined as the rate of doing ______.

Magic Solution: Power \(P = W/t\), which is the rate at which work is performed.
MODERATE

5. An object of mass 2kg is moving with a velocity of 4m/s. Its Kinetic Energy is:

Magic Solution: \(E_k = \frac{1}{2} \times 2 \times 4^2 = 1 \times 16 = 16 \, J\).
MODERATE

6. How much work is done by a girl weighing 40kg climbing a 3m high ladder? (\(g=10m/s^2\))

Magic Solution: \(W = mgh = 40 \times 10 \times 3 = 1200 \, J\).
MODERATE

7. If the velocity of a car is doubled, its Kinetic Energy becomes:

Magic Solution: Energy is proportional to \(v^2\). If \(v \times 2\), then \(v^2 \times 4\).
MODERATE

8. A machine does 500J of work in 10 seconds. What is its Power?

Magic Solution: \(P = W/t = 500/10 = 50 \, W\).
COMPLEX

9. A bulb of 100W is used for 10 hours daily. Units of energy consumed in one day is:

Magic Solution: Energy = \((100W \times 10h) / 1000 = 1 \, kWh = 1 \, Unit\).
COMPLEX

10. When an object falls freely towards the ground, its Total Mechanical Energy:

Magic Solution: According to the Law of Conservation of Energy, Total Energy (\(E_k + E_p\)) remains constant.

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