Source: Final Exam Formula Booklet
Tags: kinematics, linear momentum, kinetic energy, potential energy, friction, elastic energy, UAE, universal accounting equation, conservation of energy, ENGR 216, PHYS 216, Texas A&M
This covers the core mechanics toolkit: kinematic equations for constant acceleration, momentum, kinetic and potential energy, friction, and the Universal Accounting Equation (UAE). These formulas appear in nearly every numeric workout problem on the exam, either on their own or combined with rotational dynamics.
Linear momentum (p)
The product of mass and velocity: p = mv. A vector quantity, same direction as v.
Kinetic energy (KE)
The energy of motion: KE = (1/2)mv². A scalar, always non-negative.
Gravitational potential energy (U_grav)
Energy due to position in a gravitational field: U_grav = mgh. The reference height (h = 0) is your choice, but it must stay consistent within a problem.
Elastic potential energy (U_elastic)
Energy stored in a deformed spring: U_elastic = (1/2)kx². x is the displacement from the spring's natural (unstretched) length.
Kinetic friction (f_k)
The friction force on a sliding object: f_k = μ_k F_N. Acts opposite to the direction of motion.
Static friction (f_s)
The friction force that prevents sliding: f_s ≤ μ_s F_N. The formula gives the maximum value. Static friction adjusts to match the applied force up to that limit.
Universal Accounting Equation (UAE)
A general balance principle: Final Amount − Initial Amount = Input − Output + Generation − Consumption. Applies to mass, energy, momentum, and other conserved (or tracked) quantities.
These apply when acceleration is constant along a straight line.
s(t) = sᵢ + vᵢt + (1/2)at²
v_f² = vᵢ² + 2a(s_f − sᵢ)
Where s is position, v is velocity, a is acceleration, and t is time.
The first equation gives position as a function of time. The second eliminates time, which is useful when time is not given or asked for.
p = mv (vector)
Conservation of momentum applies when the net external force on a system is zero: Σp_before = Σp_after
Kinetic energy:
Translational: KE = (1/2)mv²
Rotational: KE = (1/2)Iω²
Total (rolling object): KE = (1/2)mv² + (1/2)Iω²
Potential energy:
Gravitational: U = mgh
Elastic (spring): U = (1/2)kx²
Conservation of energy (no non-conservative forces doing work):
KE_i + U_i = KE_f + U_f
When friction or other non-conservative forces are present, their work equals the change in mechanical energy.
Kinetic: f_k = μ_k F_N (object is sliding)
Static: f_s ≤ μ_s F_N (object is not sliding, formula gives maximum)
F_N is the normal force, which is not always equal to mg (it depends on the situation, e.g. inclined planes, applied forces)
Final − Initial = Input − Output + Generation − Consumption
This is a bookkeeping framework. For conserved quantities (energy in a closed system, mass in a non-reactive system), Generation and Consumption are zero and it reduces to:
Final − Initial = Input − Output
For an isolated system (no input or output), it further reduces to Final = Initial, which is just a conservation law.
g = 9.8 m/s²
1 revolution = 2π radians
Quantity | Formula |
|---|---|
Position (const. accel.) | s = sᵢ + vᵢt + (1/2)at² |
Velocity-displacement | v_f² = vᵢ² + 2a(s_f − sᵢ) |
Linear momentum | p = mv |
Translational KE | (1/2)mv² |
Rotational KE | (1/2)Iω² |
Gravitational PE | mgh |
Elastic PE | (1/2)kx² |
Kinetic friction | f_k = μ_k F_N |
Static friction (max) | f_s = μ_s F_N |
UAE | Final − Initial = In − Out + Gen − Consumption |
⚠️ The kinematic equations only work for constant acceleration. If acceleration varies, you need calculus or energy methods.
⚠️ Normal force F_N is not always mg. On an incline, F_N = mg cos θ. With an applied vertical force, it changes accordingly.
⚠️ Static friction is self-adjusting up to its maximum. The formula f_s = μ_s F_N gives the maximum, not the value in every situation. Many problems ask for the threshold where sliding begins.
⚠️ When combining translational and rotational KE (e.g. a ball rolling down a ramp), remember to use the appropriate moment of inertia and the constraint v = rω for rolling without slipping.
⚠️ The UAE is a general framework. Exam problems may ask you to identify which terms are zero for a given scenario (e.g. no generation or consumption for conserved quantities, no input or output for an isolated system).
⚠️ The exam says: do not include units in numeric answers, do not use scientific notation, and round only the final answer.
Q: A 2 kg object starts from rest and accelerates at 3 m/s² for 4 seconds. How far does it travel?
A: s = 0 + 0(4) + (1/2)(3)(4²) = 24 m.
Q: The same object from the previous question: what is its final velocity?
A: v_f² = 0 + 2(3)(24) = 144, so v_f = 12 m/s. (Or simply v = vᵢ + at = 0 + 3(4) = 12 m/s.)
Q: A 5 kg block slides on a surface with μ_k = 0.3. The block is on a level surface. What is the friction force?
A: F_N = mg = 5(9.8) = 49 N. f_k = 0.3(49) = 14.7 N.
Q: A spring with k = 500 N/m is compressed 0.08 m. How much elastic PE is stored?
A: U = (1/2)(500)(0.08²) = (1/2)(500)(0.0064) = 1.6 J.
Q: State the UAE for an isolated system with no generation or consumption.
A: Final Amount = Initial Amount. This is a conservation law.
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