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Flashcards · AP® Physics C: Mechanics · Unit 2

Unit 2: Force and Translational Dynamics flashcards

78 cards on topics 2.1 to 2.10. Pick a topic or study the whole unit.

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2.1 Systems and Center of Mass

What is a system in mechanics?
Whatever you put inside a chosen boundary; everything else is the environment. A system's behaviour comes from the interactions between its parts, and mass or energy may cross the boundary in an open system.
When can a system be modelled as a single object?
When its internal structure does not matter to the question. Then treat it as one object located at its center of mass. If parts move differently and that matters (water surging in a braking tank), you cannot.
How do you find the center of mass of a set of particles?
x_cm = Σmᵢxᵢ / Σmᵢ, and the same for y and z. Divide by the total mass, not the number of particles, and keep the sign of negative coordinates.
How do you find the center of mass of a continuous object?
x_cm = ∫x dm / ∫dm. For a rod, write dm = λ dx where λ = dm/dx is the linear density in kg/m, so x_cm = ∫xλ dx / ∫λ dx.
How do you find the mass of a rod whose density varies?
M = ∫λ dx over its length. Use λ × length only when λ is constant. Plates use M = ∫σ dA (σ in kg/m²) and solids use M = ∫ρ dV (ρ in kg/m³).
Must the center of mass lie on the object?
No. A uniform ring has its center of mass at its center, where there is no material. Also, the center of mass does not split the mass in half on each side.
How do you find the center of mass of a shape with a hole?
Treat the hole as a piece of negative mass at its own center, combined with the full shape using x_cm = Σmᵢxᵢ / Σmᵢ.
A 2.0 m rod has λ = 3.0x (kg/m, x from one end). Find M and x_cm.
M = ∫₀² 3.0x dx = 1.5x² = 6.0 kg. ∫x dm = ∫₀² 3.0x² dx = 8.0 kg·m. x_cm = 8.0 ÷ 6.0 = 4/3 m ≈ 1.3 m from the light end.

2.2 Forces and Free-Body Diagrams

What is a force?
A vector describing an interaction between two objects or systems. Name both: "tension, cable on crate". An object cannot exert a net force on itself; internal forces cancel in pairs.
What is a free-body diagram, and how should it be drawn?
It shows only the forces exerted on the chosen system by its environment. Use a dot for the center of mass and one straight arrow per force starting on the dot, labelled "by what, on what". No components and no merged arrows.
Why should "ma" or "force of motion" never appear on a free-body diagram?
Every force must have a "by what". ma is the result of the forces, not a force, and "force of motion" or "inertia" has no object exerting it.
What are contact forces at the atomic level?
Normal force, friction, tension, spring and drag forces are the large-scale effect of electric forces between atoms. Long-range forces such as gravity need no contact.
What are the components of gravity on a slope of angle θ?
mg sin θ down the slope and mg cos θ into the slope. Choose one axis along the slope so the motion is along one axis.
Is the normal force always equal to mg?
No. Find it from the free-body diagram. Angled pulls, extra pushes, slopes and accelerating lifts all change it. A 5.0 N push down on a 2.0 kg book gives N = 19.6 + 5.0 ≈ 25 N.
What exerts the forward force on a sprinter leaving the blocks?
The blocks and ground push forward on her feet. Her muscles are internal to the system "sprinter" and cannot accelerate her center of mass on their own.

2.3 Newton’s Third Law

What does Newton's third law state?
If A exerts a force on B, then B exerts a force on A at the same instant, equal in size and opposite in direction: F_B on A = −F_A on B. The two forces are the same type and act on different objects.
Are an object's weight and the normal force on it a third-law pair?
No. Both act on the same object and are different types. The partner of Earth's pull on the book is the book's pull on Earth; the partner of the table's push on the book is the book's push on the table.
Why can third-law partners never cancel on one free-body diagram?
They act on different objects, so they appear on two separate diagrams. Only forces on the same object can be added together.
Why do internal forces not change the motion of a system's center of mass?
Forces between parts of the system come in third-law pairs that add to zero, so ΣF over all parts equals ΣF_ext. Two skaters pushing apart on smooth ice leave their center of mass at rest.
How does tension vary along a heavy rope hanging at rest?
dT/dy = λg, with y measured up from the bottom, so T(y) = T(0) + ∫₀ʸ λg dy′. Tension is largest at the top, where it supports the whole rope below.
What makes a string or pulley "ideal"?
An ideal string has negligible mass (λ → 0) and does not stretch, so T is the same everywhere. An ideal pulley is massless with a frictionless axle, so the tension is the same on both sides.
Does the bigger object exert the bigger force in a collision?
No. The two forces of an interaction are always equal in size. The accelerations differ because the masses differ.
A uniform 4.0 m rope of mass 2.0 kg hangs from a beam. What is the tension 1.0 m below the top?
That point supports the 3.0 m of rope below it, mass 1.5 kg. T = 1.5 × 9.8 ≈ 15 N.

2.4 Newton’s First Law

What does Newton's first law state?
If the net force on a system is zero, its velocity stays constant in both size and direction. Being at rest is one case of constant velocity.
What is translational equilibrium?
ΣF = 0, applied as separate component equations ΣF_x = 0 and ΣF_y = 0 (and ΣF_z = 0). In two dimensions this lets you find up to two unknowns.
Is an object momentarily at rest in equilibrium?
Not necessarily. Check the forces, not the velocity. A ball at the top of a throw has v = 0 but ΣF = mg downward.
Is an object moving at constant speed in a circle in equilibrium?
No. Its direction is changing, so its velocity is changing and ΣF ≠ 0.
What is an inertial reference frame?
A frame in which the first law is observed to hold. Any frame moving at constant velocity relative to an inertial frame is also inertial; accelerating frames are not. The ground is usually treated as inertial.
Can forces balance along one axis but not another?
Yes. If ΣF_x = 0, v_x stays constant even while v_y changes. For x = 4.0t and y = 2.0 + 1.5t², forces balance along x only.
Does motion need a force to keep going?
No. Only a change in velocity needs a net force. A crate pushed at constant velocity has friction equal and opposite to the push, so ΣF = 0.

2.5 Newton’s Second Law

What does Newton's second law state for a system?
a_cm = ΣF / m_sys. The acceleration of the center of mass is proportional to the net external force, inversely proportional to the system's mass, and in the direction of the net force.
How do you apply Newton's second law in components?
Write ΣF_x = ma_x and ΣF_y = ma_y separately, one equation per axis, including every external force with its sign.
How do you find velocity when the net force varies with time?
a_x(t) = ΣF_x(t)/m at every instant, so v_x(t) = v_x0 + (1/m)∫₀ᵗ ΣF_x dt. The change in v_x equals the area under the ΣF_x–t graph divided by m.
What is a newton in base units?
1 N = 1 kg·m/s², the net force that gives a 1 kg mass an acceleration of 1 m/s².
How can a lab test Newton's second law with graphs?
Plot a against ΣF at fixed mass: a straight line through the origin with slope 1/m. Or plot a against 1/m at fixed net force: a straight line with slope ΣF.
Does the net force set the velocity?
No. The net force sets the acceleration. The velocity depends on all earlier forces, so velocity and net force can point in different directions.
Which mass goes in a = ΣF/m for connected objects?
The mass of the system you chose. If the system contains both objects, the forces between them are internal and do not appear in ΣF.
A 2.0 kg cart starts from rest with net force F_x = (6.0 N/s)t. What is v_x at 2.0 s?
a_x = F_x/m = 3.0t m/s². v_x = ∫₀ᵗ 3.0t dt = 1.5t². At 2.0 s, v_x = 6.0 m/s.

2.6 Gravitational Force

What is Newton's law of universal gravitation?
F_g = G m₁m₂ / r², attractive, along the line between centers, with G = 6.67 × 10⁻¹¹ N·m²/kg² and r the center-to-center distance. Each object pulls the other with the same size of force.
What is the gravitational field g?
Force per unit mass: g = F_g/m = GM/r², pointing toward M, in N/kg (equal to m/s²). With gravity as the only force, an object's acceleration is g.
What is apparent weight?
The normal force from the support. With +y up, N = m(g + a_y). It depends on acceleration, not velocity. "Weightless" means N = 0: gravity may still act.
What is the gravitational force inside a thin spherical shell?
Zero everywhere inside. Outside, the shell acts as if its mass were at its center: F = GMm/r².
What is the gravitational force inside a uniform solid sphere?
Only the mass within r counts: M_partial = Mr³/R³, so F = (GMm/R³)r. The force is proportional to r and points toward the center, F = −kr.
How does surface gravity depend on a uniform planet's size and density?
g_surface = (4/3)πGρR. With the same density, doubling the radius doubles g.
What is the equivalence principle?
Inside a closed cabin you cannot tell a gravitational field from an acceleration of the cabin. Experiments also show inertial mass and gravitational mass are equal.
What is a common mistake with r in F = Gm₁m₂/r²?
Using the altitude as r. Measure r from the planet's center, so add the planet's radius. Also remember to square r.

2.7 Kinetic and Static Friction

What is the difference between static and kinetic friction?
Kinetic friction acts when surfaces slide: F_f,k = μ_k F_N. Static friction acts when they do not slide and takes whatever size is needed, up to |F_f,s| ≤ μ_s F_N. Usually μ_s > μ_k; both have no unit.
When can you write F_f,s = μ_s F_N?
Only when slipping is just about to start. Otherwise static friction is less than its maximum and is found from Newton's laws.
How do you test whether an object slips?
Assume it does not slip, find the friction force needed for that, then compare it with μ_s F_N. If the required friction exceeds μ_s F_N, it slips and kinetic friction applies.
Does friction always oppose motion?
Friction opposes relative motion between the surfaces. It can speed an object up, for example static friction from a moving belt or a floor accelerates a crate or a walker.
What is the condition for a block to rest on a slope?
tan θ ≤ μ_s. The mass cancels because both the down-slope component mg sin θ and the maximum friction μ_s mg cos θ are proportional to m.
What is the acceleration of a block sliding down a rough slope?
With +x down the slope, a_x = g(sin θ − μ_k cos θ), using F_N = mg cos θ when no other force acts perpendicular to the slope.
Does friction depend on the area of contact?
No. In the course model friction depends only on the normal force and the coefficient for the two materials.
A 1.0 kg block on a level floor (μ_s = 0.50) is pushed with P = (3.0 N/s)t. When does it start to slide?
It slides when P reaches μ_s mg = 0.50 × 1.0 × 9.8 = 4.9 N, so t = 4.9 ÷ 3.0 ≈ 1.6 s.

2.8 Spring Forces

What is Hooke's law for an ideal spring?
F_s,x = −kx, where x is the change in length from the relaxed length and k is the spring constant in N/m. The force points back toward the relaxed length: a stretched spring pulls, a compressed spring pushes.
What makes a spring ideal?
It has negligible mass and exerts a force exactly proportional to its change in length. A massless spring pulls with the same force at both ends. A spring with real mass or a curved force–extension graph is nonideal.
How do you find k from a force–extension graph?
k is the slope of the force against extension graph. If the graph curves, the spring is nonideal; use the local slope rather than F = kx.
How do springs combine in series and in parallel?
Series (same force): 1/k_eq = 1/k₁ + 1/k₂ + …, softer than the softest spring. Parallel (same extension): k_eq = k₁ + k₂ + …. Ask "same force or same extension?", not how it looks.
What is the equilibrium stretch of a block hanging from a spring?
s₀ = mg/k. At this stretch the spring force balances the weight, and the net force on a displaced block points back toward this equilibrium.
What is the spring constant of one piece of a cut spring?
A spring cut into n equal pieces gives pieces of constant nk each. Shorter pieces are stiffer.
What unit mistake is common with springs?
Using centimetres with k in N/m. Convert to metres first: 2.5 cm compression of a 400 N/m spring gives 400 × 0.025 = 10 N.

2.9 Resistive Forces

What resistive force model does the course use?
F_r = −kv, proportional to velocity and opposite to it, with k in kg/s. It is a model: drag on fast objects is closer to proportional to v².
How do you solve Newton's second law with a resistive force?
m dv/dt = ΣF has v on both sides. Separate variables (v terms with dv, t terms with dt), then integrate from the initial state (0, v₀) to a general state (t, v).
What is the velocity of an object coasting with drag F = −kv?
m dv/dt = −kv gives v = v₀ e^(−t/τ) with τ = m/k. v → 0 and the total distance approaches mv₀/k = v₀τ.
What is the velocity of an object falling from rest with drag F = −kv?
With +y down, m dv/dt = mg − kv gives v = v_T(1 − e^(−t/τ)), with v_T = mg/k and τ = m/k. The acceleration is a = g e^(−t/τ).
What is terminal velocity?
The velocity at which the drag balances the constant force, so ΣF = 0. With F = −kv falling, v_T = mg/k. Two forces still act; only the net force is zero.
How complete is an exponential approach after one or five time constants?
After one τ it is 63% complete; after 5τ it is over 99% complete. In coasting, the time to halve the speed is τ ln 2.
Why can't the constant-acceleration equations be used with drag?
The acceleration depends on v, so it changes as the object speeds up or slows down. Integrate the differential equation instead.
What is the acceleration of an object released from rest with linear drag?
At release v = 0, so there is no drag and a = g. As v → v_T, a → 0. At half the terminal speed, drag = mg/2, so a = g/2.

2.10 Circular Motion

Why is an object in uniform circular motion accelerating?
The direction of its velocity changes even at constant speed. Differentiating r(t) = r cos(ωt) î + r sin(ωt) ĵ twice gives a = −ω²r: size v²/r, toward the center.
What are the formulas for centripetal acceleration?
a_c = v²/r toward the center, on any circular path. At constant speed, also a_c = 4π²r/T², using T = 2πr/v.
What is tangential acceleration, and how does it combine with centripetal acceleration?
a_t = dv/dt, along the velocity, nonzero only when the speed changes. It is perpendicular to a_c, so the total is |a| = √(a_c² + a_t²).
How do you apply Newton's second law to circular motion?
Take + toward the center and write ΣF toward the center = mv²/r. "Centripetal force" is this inward net force, never an extra arrow on the free-body diagram.
What is the minimum speed at the top of a vertical loop?
v_min = √(gr), when the normal force (or tension) at the top just reaches zero and gravity alone provides the centripetal force.
At what speed does a banked curve need no friction?
v² = rg tan θ, where θ is the bank angle. Faster than this, friction acts down the slope; slower, it acts up the slope.
What are the results for a conical pendulum?
tan θ = v²/(rg), and the period is T = 2π√(L cos θ/g), where L is the string length and θ its angle from the vertical.
What is Kepler's third law for a circular orbit?
T² = (4π²/GM) r³, found by setting GMm/r² = mv²/r with v = 2πr/T. r is the orbit radius from the planet's center.
A ball moves in a circle of radius 0.50 m at 2.0 Hz. Find T, v and a_c.
T = 1/f = 0.50 s. v = 2πrf = 2π × 0.50 × 2.0 ≈ 6.3 m/s. a_c = v²/r ≈ 79 m/s².

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