Summit Olympiad
Physics Resources
Work, Energy & Power Guide
Summit Olympiad Resource Library
F=ma Exam Focus
This guide is tailored for the F=ma Exam, focusing on the conservation laws that
simplify complex motion problems. It covers the Work-Energy Theorem, Conservative
vs. Non-Conservative forces, Hooke’s Law (Springs), and Power. Mastery of these scalar
quantities is essential for competition physics.
Contents
1 Work (W )
1.1 The Definition of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Work-Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2 Mechanical Energy
2.1 Kinetic Energy (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Potential Energy (U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Conservative vs. Non-Conservative Forces . . . . . . . . . . . . . . . . . . . . . .
2
2
2
2
3 Conservation of Energy
3.0.1 Practice Problems: Conservation Strategies . . . . . . . . . . . . . . . . .
2
3
4 Power (P )
4.1 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Practice Problems: Power . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
1
Work (W )
Work is the transfer of energy into or out of a system by a force acting on an object over a
displacement. Unlike Velocity or Force, Work is a scalar quantity.
1.1
The Definition of Work
For a constant force, work is defined as the dot product of the Force vector and Displacement
vector.
W = F⃗ · d⃗ = F d cos(θ)
• θ: The angle between the force vector and the direction of motion.
• Positive Work (0 ≤ θ < 90◦ ): The force helps the motion (adds energy).
Reach Your Academic Summit
Page 1
Summit Olympiad
Physics Resources
• Negative Work (90◦ < θ ≤ 180◦ ): The force opposes motion (removes energy, e.g.,
Friction).
• Zero Work (θ = 90◦ ): Forces perpendicular to motion (like Normal Force or Centripetal
Force) do no work.
1.2
The Work-Energy Theorem
This is the bridge between Newton’s Laws and Energy. It states that the Net Work done on
an object equals its change in Kinetic Energy.
Wnet = ∆K = Kf − Ki
2
Mechanical Energy
Energy is the capacity to do work. We categorize mechanical energy into two main forms.
2.1
Kinetic Energy (K)
The energy of motion. It is always non-negative.
1
K = mv 2
2
2.2
Potential Energy (U )
Potential energy is ”stored” energy due to an object’s position. It is only defined for Conservative Forces.
1. Gravitational Potential Energy (Ug ): For objects near Earth’s surface (where g is
constant):
Ug = mgh
2. Elastic Potential Energy (Us ): Stored in ideal springs that obey Hooke’s Law (Fs =
−kx).
1
Us = kx2
2
where k is the spring constant (N/m) and x is the stretch/compression distance from
equilibrium.
2.3
Conservative vs. Non-Conservative Forces
• Conservative Forces (Gravity, Springs): Work done is path independent. Mechanical energy is conserved if only these forces do work.
• Non-Conservative Forces (Friction, Air Resistance): Work done depends on the
path. These forces dissipate mechanical energy into heat (Ethermal ).
3
Conservation of Energy
If the work done by non-conservative forces (Wnc ) is zero, the total mechanical energy of the
system remains constant.
Ei = Ef
Ki + Ug,i + Us,i = Kf + Ug,f + Us,f
Reach Your Academic Summit
Page 2
Summit Olympiad
3.0.1
Physics Resources
Practice Problems: Conservation Strategies
Problem 1: A 2 kg block slides down a frictionless curved ramp starting from rest at a height
of 5 m. At the bottom, it hits a spring with k = 400 N/m. How far does the spring compress?
Solution: Identify the energy transfer: Gravitational PE → Kinetic Energy → Elastic PE.
We can skip the middle step and equate the initial and final states.
Initial State (Top): v = 0, spring is relaxed (x = 0), height h = 5.
Ei = mgh = (2)(10)(5) = 100 J
Final State (Max Compression): v = 0 (momentarily stopped), height h = 0, spring
compressed x.
1
1
Ef = kx2 = (400)x2 = 200x2
2
2
Set Ei = Ef :
100 = 200x2 =⇒ x2 = 0.5 =⇒ x ≈ 0.707 m
Problem 2: A pendulum bob of mass m is released from an angle where it is 0.8 m higher
than its lowest point. What is its speed at the lowest point?
Solution: Tension force is perpendicular to motion, so it does zero work. Only gravity
acts, so Energy is conserved.
Utop = Kbottom
1
mgh = mv 2
2
Mass cancels out (classic F=ma trick).
p
p
√
v = 2gh = 2(10)(0.8) = 16 = 4 m/s
4
Power (P )
Power measures how fast work is done. It is crucial for problems involving engines, motors, or
timed lifts.
4.1
Formulas
• Average Power:
Pavg =
∆E
W
=
∆t
∆t
• Instantaneous Power: (Useful when force and velocity are known)
P = F⃗ · ⃗v = F v cos(θ)
Unit: The Watt (W). 1 W = 1 J/s.
Reach Your Academic Summit
Page 3
Summit Olympiad
4.1.1
Physics Resources
Practice Problems: Power
Problem 3: A 1000 kg car accelerates from 0 to 20 m/s in 5 seconds on a flat road. Neglecting
friction, what is the average power output of the engine?
Solution: First, calculate the Work done. By the Work-Energy Theorem, Work equals
the change in Kinetic Energy.
1
1
W = ∆K = mvf2 − mvi2
2
2
1
W = (1000)(20)2 − 0 = 500(400) = 200, 000 J
2
Now, find Power:
P =
W
200, 000
=
= 40, 000 W = 40 kW
t
5
Problem 4: A cyclist is riding up a 30◦ hill at a constant speed of 6 m/s. If the combined mass
of the cyclist and bike is 80 kg, how much power must they generate to maintain this speed?
(Ignore air resistance)
Solution: Since speed is constant, the net force is zero. The cyclist’s applied force must
exactly counter the component of gravity pulling them down the slope.
Fgravity,parallel = mg sin(30◦ )
Fapplied = (80)(10)(0.5) = 400 N
Use the Instantaneous Power formula (P = F v):
P = (400 N)(6 m/s) = 2400 W
To track progress, and to check what you’ve learned, sign in to the Summit Olympiad
Dashboard.
Reach Your Academic Summit
Page 4