Lesson 10 of 10 · 17 min
Chapter review
Must-know facts
18 facts
- 1W = F·d = Fd cos θ; zero when force and displacement are perpendicular.
- 2Work by friction on a sliding body and by gravity on a rising body is negative.
- 31 kWh = 3.6 × 10⁶ J; 1 eV = 1.6 × 10⁻¹⁹ J; 1 erg = 10⁻⁷ J; 1 hp = 746 W.
- 4Work-energy theorem: W_net = ΔK, valid for constant and variable forces.
- 5K = ½mv² = p²/2m; K ∝ p² for fixed mass.
- 6A rise of x% in momentum raises K by ((1 + x/100)² − 1) × 100%.
- 7Work by a variable force = area under the F-x graph.
- 8Conservative force: path-independent work, zero work round a closed path; F = −dV/dx.
- 9Friction and air resistance are non-conservative.
- 10Gravitational PE near the surface = mgh; spring PE = ½kx².
- 11Mechanical energy is conserved only when non-conservative forces do no work.
- 12Stopping distance on a rough floor d = v²/(2μ_k g), independent of mass.
- 13Vertical circle on a string: minimum speed √(5gL) at the bottom and √(gL) at the top.
- 14P = dW/dt = F·v.
- 15Momentum is conserved in all collisions; kinetic energy only in elastic ones.
- 16Equal masses in a 1D elastic collision exchange velocities.
- 17Fraction of KE transferred by m₁ to m₂ (at rest) in a 1D elastic collision = 4m₁m₂/(m₁ + m₂)².
- 18Perfectly inelastic collision gives the largest KE loss; bodies move together afterwards.
Common traps
Where marks are lost
Writing W = Fd when the force acts at an angle to the displacement.
Using mgh = ½mv² when part of the path is rough.
Assuming that if momentum rises by 30% then kinetic energy rises by 30%.
Believing each body keeps its own kinetic energy in an elastic collision.
Thinking kinetic energy is conserved at every instant during an elastic collision.
Treating the kilowatt hour as a unit of power.
Forgetting efficiency in pump or motor questions, or dividing the wrong way.
Assigning potential energy to friction.
Taking the potential energy of a spring as ½kx when finding its maximum compression.
Formulas
16 to know
Scalar product
A·B = AB cos θ = AxBx + AyBy + AzBz
θ is the angle between A and B.
Work by a constant force
W = F·d = Fd cos θ
SI unit joule (N m).
Work by a variable force
W = ∫ F(x) dx from x_i to x_f
Area under the F-x graph.
Kinetic energy
K = ½mv² = p²/2m
Always ≥ 0.
Work-energy theorem
K_f − K_i = W_net
W_net includes work by every force, conservative or not.
Conservative force from potential energy
F(x) = −dV/dx
One-dimensional form.
Gravitational potential energy near the surface
V(h) = mgh
h measured from the chosen zero level.
Conservation of mechanical energy
K_i + V_i = K_f + V_f
Only when conservative forces alone do work.
Spring force (Hooke's law)
F_s = −kx
k in N m⁻¹.
Spring potential energy
V(x) = ½kx²
Same for extension and compression x.
Work by a spring
W_s = −½k(x_f² − x_i²)
Negative when the spring is stretched further.
Power
P_av = W/t; P = dW/dt = F·v
1 W = 1 J s⁻¹; 1 hp = 746 W.
Efficiency
η = useful output power / input power
Always less than 1 in practice.
1D elastic collision, m₂ initially at rest
v₁f = (m₁ − m₂)v₁ᵢ/(m₁ + m₂); v₂f = 2m₁v₁ᵢ/(m₁ + m₂)
Momentum and total KE both conserved.
Completely inelastic collision, m₂ at rest
v = m₁v₁ᵢ/(m₁ + m₂); ΔK = ½[m₁m₂/(m₁ + m₂)]v₁ᵢ²
Bodies stick together; ΔK is the kinetic energy lost.
Vertical circle on a string
v_bottom(min) = √(5gL); v_top(min) = √(gL)
L is the string length.
Key terms
14 terms
- Scalar product
- A product of two vectors that gives a scalar, AB cos θ.
- Work
- The product of a force and the component of displacement along it.
- Joule
- The work done when a force of 1 N moves its point of application 1 m along the force.
- Kinetic energy
- The energy a body has because it is moving, ½mv².
- Work-energy theorem
- The net work done on a body equals the change in its kinetic energy.
- Potential energy
- Stored energy that depends on the position or configuration of a system.
- Conservative force
- A force whose work between two points does not depend on the path taken.
- Non-conservative force
- A force, such as friction, whose work depends on the path and which reduces mechanical energy.
- Mechanical energy
- Kinetic energy plus potential energy of a system.
- Spring constant
- The force per unit extension of a spring; a measure of its stiffness.
- Power
- Work done, or energy transferred, per unit time.
- Elastic collision
- A collision in which total kinetic energy after impact equals that before.
- Inelastic collision
- A collision in which some kinetic energy is converted to other forms.
- Completely inelastic collision
- An inelastic collision in which the bodies move together after impact.