10.1 Mechanics

10.1 Mechanics on NMAT Physics

The Center for Educational Measurement (CEM) NMAT Physics subtest is a 30-item block with a recommended ~30 minutes. Official content areas include Mechanics, Thermodynamics, Vibrations, Waves & Optics, Electricity & Magnetism, and Modern Physics. This section covers Mechanics — the largest “classical” slice and a high-yield source of formula-application items.

Items demand understanding, applying, analyzing, evaluating, and synthesizing at introductory college depth. You are expected to select the right law, keep SI units consistent, and finish multi-step arithmetic without a calculator culture of “plug and pray.”

Quick frame: Kinematics describes motion without asking why. Newton’s laws connect force to acceleration. Energy and momentum give conservation shortcuts. Fluids reuse density and pressure ideas that also appear in physiology contexts later in medical training.

Core formula table (SI units)

Use (g = 9.8,\mathrm{m/s^2}) when precision is asked; many NMAT-style estimates allow (g = 10,\mathrm{m/s^2}) if options are widely spaced — state the choice and stay consistent.

Kinematics: velocity, acceleration, free fall

Position (x), displacement (\Delta x), velocity (v), and acceleration (a) are vector ideas in 1D (sign encodes direction). The constant-acceleration equations hold only when (a) is constant (including free fall with constant (g), neglecting air resistance).

Free fall: take upward positive → (a = -g); take downward positive → (a = +g). Pick one convention per problem. At the top of a vertical toss, (v = 0) but (a = -g) still (acceleration is not zero at the apex).

Worked example — free fall time. A stone is dropped from rest from a height of 45 m. How long to hit the ground? Use (g = 10,\mathrm{m/s^2}), downward positive.

• (y = v_0 t + \tfrac{1}{2}at^2 \Rightarrow 45 = 0 + \tfrac{1}{2}(10)t^2 \Rightarrow 45 = 5t^2 \Rightarrow t^2 = 9 \Rightarrow t = 3,\mathrm{s}).
• Impact speed: (v = 0 + (10)(3) = 30,\mathrm{m/s}) (or (v^2 = 2a\Delta y = 2(10)(45) = 900 \Rightarrow v = 30,\mathrm{m/s})).

Horizontal projectile: (a_x = 0), (a_y = -g). Time of flight comes from the vertical equation; range uses that time with constant (v_x).

Newton’s laws and friction intro

1. 1st law (inertia): If (\sum F = 0), velocity is constant (including rest). “No net force” ≠ “no forces.”
2. 2nd law: (\sum \vec{F} = m\vec{a}). Draw a free-body diagram; resolve axes along/against motion or along the incline.
3. 3rd law: Action–reaction pairs act on different objects, equal magnitude, opposite direction, same type of force.

Weight (mg) is the gravitational force; normal force (N) is perpendicular contact force — not always equal to weight (accelerating elevators, inclines, stacked objects).

Friction intro. Kinetic friction often (f_k = \mu_k N), opposes relative sliding. Static friction (f_s \le \mu_s N) adjusts up to a maximum to prevent slipping. On a level surface with no other vertical forces, (N = mg); on an incline of angle (\theta), (N = mg\cos\theta) and the parallel component is (mg\sin\theta).

Worked example — Newton on a level surface. A 5.0 kg crate is pulled horizontally by 20 N while kinetic friction is 8.0 N. Find acceleration.

• (\sum F_x = 20 - 8 = 12,\mathrm{N}); (a = F/m = 12/5 = 2.4,\mathrm{m/s^2}).

Work, energy, power, and conservation

Work by a constant force: (W = F\Delta x\cos\theta). Positive work adds energy to the system (as defined); negative work removes it. The work–energy theorem: net work on a particle equals (\Delta K).

Mechanical energy (E = K + U). If only conservative forces do work (or nonconservative work is zero), (E) is conserved: (K_i + U_i = K_f + U_f). If friction does work (W_{\mathrm{nc}}), then (K_i + U_i + W_{\mathrm{nc}} = K_f + U_f) (with (W_{\mathrm{nc}}) negative for kinetic friction).

Power is the rate of energy transfer: (P_{\mathrm{avg}} = W/\Delta t); instantaneous along motion (P = Fv).

Worked example — energy conservation. A 2.0 kg block slides from rest down a frictionless track from height 5.0 m. Speed at the bottom?

• (mgh = \tfrac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2(9.8)(5)} = \sqrt{98} \approx 9.9,\mathrm{m/s}).
• Mass cancels — a common NMAT insight when options depend only on (h).

Worked example — power. A motor does 6000 J of work in 4.0 s. Average power = (6000/4 = 1500,\mathrm{W} = 1.5,\mathrm{kW}).

Momentum and collisions

Linear momentum (\vec{p} = m\vec{v}). Impulse (\vec{J} = \vec{F}_{\mathrm{avg}}\Delta t = \Delta\vec{p}). For an isolated system (no external net impulse), total momentum is conserved.

• Elastic collision: momentum and kinetic energy both conserved (ideal billiard-ball limit).
• Inelastic collision: momentum conserved; kinetic energy not fully conserved.
• Perfectly inelastic: objects stick; maximize KE loss consistent with momentum conservation.

Worked example — 1D inelastic stick. Mass 3.0 kg at 4.0 m/s hits 1.0 kg at rest and sticks. Common speed?

• (m_1 v_1 = (m_1 + m_2)v \Rightarrow 3(4) = 4v \Rightarrow v = 3.0,\mathrm{m/s}).
• Initial (K = \tfrac{1}{2}(3)(16) = 24,\mathrm{J}); final (K = \tfrac{1}{2}(4)(9) = 18,\mathrm{J}) — KE not conserved, as expected for sticking.

Circular motion and gravity intro

Uniform circular motion needs a net centripetal force toward the center: (F_c = ma_c = mv^2/r). That force is provided by tension, gravity, friction, normal force, or a combination — not a new force type called “centrifugal” in the inertial frame.

Newton’s law of universal gravitation: (F = G m_1 m_2 / r^2). Near Earth’s surface this reduces to weight (mg) with (g = GM/R^2) for Earth. Orbital intuition: greater altitude → weaker (g); for circular orbits, gravity supplies (mv^2/r).

Worked example — centripetal force. A 0.50 kg stone on a 1.2 m string moves in a horizontal circle at 3.0 m/s (ignore gravity for the horizontal idealization). Tension (T = mv^2/r = (0.50)(9)/(1.2) = 3.75,\mathrm{N}).

Fluids: density, pressure, buoyancy

Density (\rho = m/V). Water’s density is about (1.0 \times 10^3,\mathrm{kg/m^3}) (or (1.0,\mathrm{g/cm^3})). Pressure (P = F/A). In a static fluid of constant density, pressure increases with depth: (P = P_0 + \rho g h).

Pascal’s principle: a pressure change applied to an enclosed incompressible fluid is transmitted undiminished. Hydraulic press: (F_1/A_1 = F_2/A_2), so a small force on a small piston balances a large force on a large piston (displacement is larger on the small side — work still matches in the ideal case).

Archimedes’ principle: buoyant force equals the weight of displaced fluid: (F_b = \rho_{\mathrm{fluid}} V_{\mathrm{disp}} g). An object floats when average density is less than the fluid’s; submerged fraction equals (\rho_{\mathrm{object}}/\rho_{\mathrm{fluid}}) for floating in equilibrium (simple model).

Worked example — buoyancy. A 0.80 kg wood block floats in water with 75% of its volume submerged. Find the block’s volume.

• At float: (mg = \rho_w (0.75 V) g \Rightarrow m = 0.75\rho_w V \Rightarrow V = m/(0.75\rho_w) = 0.80/(0.75 \times 1000) = 1.067 \times 10^{-3},\mathrm{m^3} \approx 1.07 \times 10^{-3},\mathrm{m^3}).

Worked example — hydrostatic pressure. Gauge pressure 5.0 m under freshwater: (\Delta P = \rho g h = (1000)(9.8)(5) = 4.9 \times 10^4,\mathrm{Pa} \approx 49,\mathrm{kPa}).

Common unit mistakes (high-yield traps)

1. Mixing g and kg in (F = ma) or (W = mg) (50 g → 0.050 kg).
2. cm and m: (h = 80,\mathrm{cm} = 0.80,\mathrm{m}); (V = 200,\mathrm{cm^3} = 2.0 \times 10^{-4},\mathrm{m^3}).
3. Using (g = 9.8,\mathrm{m/s^2}) with mass in grams without conversion.
4. Forgetting (\cos\theta) in work when force is not along displacement.
5. Confusing mass and weight on a scale reading in newtons vs kilograms.
6. Density of water: (1000,\mathrm{kg/m^3}) not (1,\mathrm{kg/m^3}); or forgetting (1,\mathrm{g/cm^3} = 1000,\mathrm{kg/m^3}).
7. Pressure vs force: large force on large area can still be modest pressure.

Exam tactics for Mechanics items

• Sketch axes and a free-body diagram before writing (\sum F = ma).
• Ask: “Is energy conserved? Is momentum conserved? Is acceleration constant?”
• Estimate with (g = 10) only when answer choices are coarse; otherwise keep 9.8.
• Cancel mass early when it appears on both sides ((v = \sqrt{2gh}), acceleration on frictionless incline (g\sin\theta)).
• For fluids, always identify which volume is displaced and which density multiplies it.

Mechanics sets the mathematical habits for the rest of Physics: define system, choose positive direction, convert units first, then compute. Thermodynamics (next section) reuses energy bookkeeping with heat and internal energy as new players.