Lesson1: Why Work Done is Equal to Change in Kinetic Energy?

1. Deriving Kinetic Energy: How do you get from Newton's Second Law (F=ma) to the Kinetic Energy formula (½mv²)?

It comes from combining Newton’s Second Law with kinematics. If you push a box with a net force F, it accelerates (a). Over a displacement d, the work done is W = Fd.

Since F = ma, then Work = mad.

Recall the kinematic equation: v_f² = v_i² + 2ad. If you solve for ad, you get ad = (v_f² - v_i²) / 2.

Substitute that back into the Work equation: W = m [ (v_f² - v_i²) / 2 ] W = ½mv_f² - ½mv_i²

This shows that the Work done equals the change in this quantity we call Kinetic Energy (K).

Lesson 2: Work Done at an Angle: Understanding the Dot Product

2. Physics "Work" vs. Real Life Work: Why is holding a heavy box stationary considered zero work in physics, even though my arms get tired?

In physics, "Work" is strictly defined as Energy Transfer via a force moving an object. The formula is W = Fd cos(θ). If the displacement d is zero, the Work is mathematically zero. Real Life: Your muscles are doing "physiological work" (contracting and twitching) to fight gravity, burning chemical energy (calories). But because the box doesn't move, you aren't transferring that energy to the box.

3. Scalar vs. Vector: Force and displacement are vectors, so why are Work and Energy scalars?

Work is the "Dot Product" of two vectors. The Dot Product multiplies the parallel parts of two vectors and results in a single number (a scalar).

Think of it like a bank account. Money enters (Positive Work) or leaves (Negative Work), but your bank balance doesn't point "North" or "East." It’s just an amount.

4. Units Confusion: What is a Joule really?

A Joule is the energy required to apply a force of 1 Newton over a distance of 1 Meter. 1 Joule = 1 N·m = 1 kg·m²/s². Comparison: A Watt is a measure of speed of energy use (Joules per second). A Joule is the "gallon of gas"; a Watt is "how fast the engine burns it."

5. The Sign of Work: What does "Negative Work" mean?

Positive work adds energy to the system (speeds it up). Negative work removes energy from the system (slows it down). Example: When you catch a fast baseball, your hand moves backward. The force you exert is forward (against the ball), but the displacement is backward. You do negative work on the ball, taking its kinetic energy away to bring it to rest.

6. Perpendicular Forces: Why is work done by the Normal Force usually zero?

The formula includes cos(θ). If Force and Displacement are perpendicular (90°), then cos(90°) = 0. Nuance: The Normal Force usually acts perpendicular to the surface you are sliding on (displacement), so it does no work. Exception: If you are in an elevator moving up, the floor pushes you up (Normal Force) and you move up (Displacement). Here, the Normal Force does positive work on you!

7. Centripetal Force: Why is the work done by a satellite in orbit exactly zero?

A centripetal force always points toward the center of the circle, while the velocity (and instantaneous displacement) points tangent to the circle. The angle between them is always exactly 90°. Therefore, no work is done, and the satellite's speed (kinetic energy) remains constant.

8. The Dot Product Angle: Which angle do I use in W = Fd cos(θ)?

Always use the angle between the Force vector and the Displacement vector when they are placed tail-to-tail. Trap: Don't blindly use the angle of the ramp or the angle with the horizontal. Look strictly at the arrow for F and the arrow for d.

Lesson 3: Work and Kinetic Energy Theorem

9. Stopping Distance Problems: How do I find stopping distance without kinematics?

Use the Work-Energy Theorem: W_net = ΔK. The only work done stopping the car is by Friction (f_k). Work = -f_k · d (Negative because friction opposes motion). Change in K = 0 - ½mv_i². So: -f_k · d = -½mv_i². Solve for d. This is often much faster than finding acceleration first.

10. Frame of Reference: Is "Work" relative?

Yes! Displacement and Velocity depend on your frame of reference. Example: If you walk on a moving train:

• Train Frame: You moved 5 meters. Work = F × 5.
• Ground Frame: The train moved 100 meters while you walked. You moved 105 meters. Work = F × 105. However, the Work-Energy theorem holds true within each consistent frame.

Lesson 4: Work Done by Gravity: Positive vs. Negative Work

11. The Gravity Sign Trap: When a ball goes up, is gravity doing positive or negative work?

• Going Up: Gravity pulls Down. Displacement is Up. They are opposite (180°). Work is Negative. (Gravity steals energy, slowing the ball).
• Coming Down: Gravity pulls Down. Displacement is Down. They are same direction (0°). Work is Positive. (Gravity adds energy, speeding the ball).
  

Lesson 5: Work Done by a Variable Force (Spring Force)

12. Variable Force Calculus: Why can't I use W = Fd for a spring?

The formula W = Fd assumes F is constant. For a spring, the force gets harder the more you stretch it (F = kx). If you use the max force, you overestimate the work. If you use the initial force (0), you underestimate it. You must use the Average Force or calculus (integration). The area under the Force vs. Position triangle is ½ × base × height = ½ · x · (kx) = ½kx².

Lesson 7: Power (P=Fv)

13. Power vs. Energy: If I do work faster, does my energy change?

The Energy (total work done) stays the same. The Power changes. Analogy: Running a mile vs. Walking a mile.

• Energy used (Work): Roughly the same (you moved your mass the same distance).
• Power: Running is High Power (done in 5 mins). Walking is Low Power (done in 15 mins).

14. Average vs. Instantaneous Power: When to use P = W/t vs P = Fv?

• Use P = W/t for "Average Power" over a long duration (e.g., "How much power did the motor average over the 10-second trip?").
• Use P = Fv for "Instantaneous Power" at a specific moment (e.g., "What is the power output exactly when the car hits 50 m/s?").
  

Lesson 8: Potential Energy vs. Work: The Negative Relation (W = -ΔU)

15. The Gradient: What does the negative sign in F(x) = -dU/dx mean?

Objects naturally "want" to go to the lowest energy state (like a ball rolling into a valley). The negative sign means the Force points in the direction where Potential Energy (U) decreases. If the slope of energy (dU/dx) is positive (uphill), the force pushes back (downhill).

Lesson 9: Conservative vs. Non-Conservative Forces

16. The Rollercoaster Loop: Does the track shape matter?

No. Gravity is a Conservative Force. The work done depends only on the starting height and ending height (Δh). The twists, loops, and length of the track are irrelevant to the final speed (assuming no friction).

17. Conservative Forces: Why is Friction non-conservative?

A force is conservative if the work done in a closed loop (returning to start) is zero.

• Gravity: Throw a ball up (-10J work) and it comes down (+10J work). Total = 0.
• Friction: Slide a block right (friction acts left, negative work). Slide it back left (friction acts right, still negative work). The energy is permanently lost to heat; you can't get it back.

Lesson 10: Deriving Potential Energy

18. Potential Energy Origin: Why is there no "Friction Potential Energy"?

Potential Energy is "Stored Energy" that can be recovered later as Kinetic Energy. When you lift a book, gravity stores that work; drop it, and you get the speed back. When you drag a block against friction, that energy turns into Heat and Sound. You cannot reverse the process to turn that heat back into block speed. Since it can't be stored, it's not Potential Energy.

Lesson 11: Law of Conservation of Mechanical Energy

19. Spring Compression: Block hits spring on frictionless surface.

Set Initial Energy = Final Energy. K_initial + U_spring-initial = K_final + U_spring-final At the moment of impact: All Kinetic (½mv²). At max compression: All Potential (½kx²) (because velocity is momentarily zero). Equation: ½mv² = ½kx². Solve for x.

20. Ramps with Friction: How to set up the equation?

The "Conservation" equation breaks because Friction "steals" energy. Use the Modified Conservation Law: E_initial - Work_friction = E_final (PE_i + KE_i) - (f_k · d) = (PE_f + KE_f). Make sure to calculate the friction distance d along the ramp, not the height.

21. Pendulum Swings: Calculating speed at the bottom.

Top of swing: Velocity = 0, Height = h. Energy = mgh. Bottom of swing: Height = 0, Velocity = v. Energy = ½mv². mgh = ½mv². Mass cancels out! v = √(2gh).

22. The "Lost" Energy: Where does it go?

Mechanical Energy is not conserved if friction is present, but Total Energy is always conserved. The "missing" Joules have turned into Internal Energy (the atoms of the block and floor vibrating faster = heat).

Lesson 12: Potential Energy Curves

23. Stability & Equilibrium: How to read the U vs. x curve?

Imagine a marble rolling on the curve.

• Stable Equilibrium: A "Valley" (U-shape). If you nudge the marble, it rolls back to the center.
• Unstable Equilibrium: A "Hill" (Inverted U). If you nudge the marble, it rolls away forever.
• Neutral Equilibrium: A flat line. Nudge it, and it stays at the new spot.
  

Lesson 13: Work Done by an External Force

24. Static Friction: Can static friction do work?

Yes! Example: You are standing on a bus that accelerates forward. What pushes you forward? Static Friction between your shoes and the floor. You move forward (Displacement) and the force is forward (Static Friction). W = F_s · d. Static friction is doing positive work on you to increase your Kinetic Energy.

25. System vs. External Work: Lifting a book.

• Work by YOU: Positive (Force up, Displacement up).
• Work by GRAVITY: Negative (Force down, Displacement up).
• Net Work: Zero (if you lift at constant speed).
• Result: The Net Work is zero, so the Kinetic Energy doesn't change (constant speed). However, your positive work was stored as Gravitational Potential Energy.