How do ideal capacitances combine in parallel and in series?

Parallel: Ceq = ΣCᵢ; voltages are equal and plate charges add. Series: 1/Ceq = Σ(1/Cᵢ); voltage drops add. Equal charge magnitudes in a series chain assume the isolated intermediate conductors have zero net charge, as for capacitors initially uncharged before connection to a battery.

How do the formulas for a parallel-plate capacitor distinguish fixed-voltage and fixed-charge changes?

Ignoring edge effects, with a uniform linear dielectric filling the gap: C = κε₀A/d, Q = CV, E = |V|/d and U = CV²/2 = Q²/(2C). A connected ideal battery fixes V, so Q and U track C. An isolated capacitor fixes Q, so V and U vary as 1/C. At fixed geometry, replacing vacuum by dielectric κ > 1 multiplies C by κ; E stays fixed with the battery connected and is divided by κ when isolated. At fixed κ and A, increasing d lowers C.

How are rotational work, power and kinetic-energy change related for a rigid body rotating about a fixed axis?

Net work is Wnet = ∫τnet dθ = Δ(Iω²/2), with constant I. Only for constant net torque does it reduce to Wnet = τnetΔθ. Instantaneous power is P = τω for torque and angular velocity components along the same fixed axis, with consistent signs.

How are moment of inertia and the parallel-axis theorem defined, and how do four standard shapes compare?

I = Σmᵢrᵢ² = ∫r² dm, with r measured perpendicular to the chosen axis. For an axis parallel to a centre-of-mass axis at separation d, I = Icm + Md². For equal M and R, uniform bodies rank: thin hoop MR² > thin spherical shell 2MR²/3 > solid disk/cylinder MR²/2 > solid sphere 2MR²/5. Use the hoop/disk axis through its centre and normal to its plane, the cylinder’s central longitudinal axis, and sphere diameters.

What defines simple harmonic motion, and what are the ideal spring and pendulum periods?

SHM obeys d²x/dt² = −ω²x, with T = 2π/ω and f = 1/T. Ideal massless linear spring: T = 2π√(m/k). Small-angle simple pendulum: T ≈ 2π√(L/g). Small-angle physical pendulum: T ≈ 2π√[I/(mgd)], where I is about the pivot and d is the pivot-to-centre-of-mass distance. Linear torsional pendulum: T = 2π√(I/κ), with restoring torque −κθ. Damping is neglected.

An ideal source ε feeds R₁, then two parallel branches: R₂ alone and R₃ in series with an initially uncharged capacitor C. What are the initial and final source currents and final capacitor charge?

For positive resistances and an ideal capacitor: at t = 0⁺, VC = 0, so I₀ = ε/[R₁ + R₂R₃/(R₂ + R₃)]. At DC steady state the R₃–C branch carries no current, giving I∞ = ε/(R₁ + R₂). With no drop across R₃, VC = εR₂/(R₁ + R₂), so Qf = CεR₂/(R₁ + R₂).

For a point at fixed radius r > 0 rotating about a fixed axis, how do angular and linear quantities relate?

With angles in radians, ω = dθ/dt and α = dω/dt. Signed arc displacement is Δs = rΔθ; tangential velocity is vₜ = rω, while speed is r|ω|. Tangential acceleration is aₜ = rα; inward radial acceleration has magnitude aᵣ = rω². The total acceleration magnitude is √(aₜ² + aᵣ²).

What are Kirchhoff’s rules, resistor-combination rules and ideal meter connections for ordinary lumped DC circuits?

At a node, total current in equals total current out. Around a loop, signed potential changes sum to zero. Series resistors add: Req = ΣRᵢ; parallel resistors obey 1/Req = Σ(1/Rᵢ). Adding a positive parallel resistance lowers Req and increases current if supply voltage is fixed. An ideal ammeter is connected in series with zero resistance; an ideal voltmeter in parallel with infinite resistance.

What are E(r) and V(r) for an isolated conducting sphere of radius R and charge Q > 0 in electrostatic equilibrium?

In vacuum, with no external fields and V(∞) = 0: inside, E = 0 and V = kQ/R; outside (r > R), E = kQ/r² radially outward and V = kQ/r. Potential is continuous at r = R. The field jumps there: its inner limit is zero and its outer limit is kQ/R².

What determines motion down an incline for a rigid body with Icm = cMR² rolling without slipping from rest?

On a stationary incline of angle θ, v = Rω and K = Mv²(1 + c)/2. With sufficient static friction and negligible rolling resistance, a = g sin θ/(1 + c); after dropping height h, v = √[2gh/(1 + c)]. Smaller c gives faster descent: solid sphere, disk, thin spherical shell, then hoop. A frictionless sliding block is faster still. Ideal static friction does no work at the stationary contact point.

What remains after a capacitor C₁ charged to V₀ is isolated from its battery and connected in parallel to an uncharged C₂?

After transients settle, charge conservation gives Vf = C₁V₀/(C₁ + C₂), Q₁ = C₁Vf and Q₂ = C₂Vf. The final stored energy is Uf = [C₁/(C₁ + C₂)]Ui. The difference becomes heat and/or radiation in a real connection; total energy is conserved. A perfectly lossless model need not settle to this static final state.

What are the moments of inertia of a uniform thin rod of mass M and length L about perpendicular axes through its centre and one end?

Through the centre: Icm = ML²/12. Through one end: Iend = Icm + M(L/2)² = ML²/3. Both axes are perpendicular to the rod; the result depends on that orientation.

An ideal constant-voltage DC source supplies R₁ in series with R₂. What happens to the power in R₁ and R₂ when a positive resistor R₃ is switched in parallel with R₂?

With fixed positive resistances, the parallel combination lowers total resistance, so current through R₁ rises and P₁ = I²R₁ increases. The larger drop across R₁ leaves less voltage across R₂, so P₂ = V₂²/R₂ decreases. This compares resistor powers; a bulb-brightness interpretation requires an appropriate lamp model.

How does a battery’s terminal voltage depend on current in the constant-emf, constant-internal-resistance model?

During discharge, Vₜ = ε − Ir. A Vₜ-versus-I graph has intercept ε and slope −r. This is a model with fixed ε and r; real batteries can change with state of charge, temperature and current.

For an ideal undamped mass–spring oscillator, what are x(t), v(t), a(t), the maxima and total energy?

Measured from equilibrium: x = A cos(ωt + φ), v = −Aω sin(ωt + φ), and a = −ω²x, with ω² = k/m. Maximum speed is Aω at x = 0; maximum acceleration magnitude is Aω² at x = ±A. Energy is E = kA²/2 = kx²/2 + mv²/2. Release from rest at x = A sets φ = 0 modulo 2π. Doubling A quadruples E but leaves the ideal period unchanged.

What are Coulomb’s law and the electric field of a stationary point charge in vacuum?

The force magnitude is F = k|q₁q₂|/r² along the line between the charges: like signs repel and unlike signs attract. Here k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C². Electric field is force per test charge, E = F/q₀ as a vector relation. A source Q gives magnitude k|Q|/r², outward for Q > 0 and inward for Q < 0; fields add as vectors.

What is torque, and what conditions keep an initially stationary rigid body in static equilibrium?

Torque is the vector cross product τ = r × F. Its magnitude is rF sin φ = Fℓ, where ℓ is the perpendicular lever arm in metres; torque is measured in N·m. Net external force and net external torque must both be zero. With force balance, torque balance holds about any fixed origin.

What are the axial field and potential of a uniformly charged thin ring of radius R and total charge Q > 0?

In vacuum, at axial coordinate z with origin at the centre: E_z = kQz/(R² + z²)^(3/2), and V = kQ/√(R² + z²), with V(∞) = 0. Transverse field components cancel. At z = 0, E = 0 but V = kQ/R. Far along the positive axis (z ≫ R), E_z ≈ kQ/z².

Where are kinetic and elastic potential energies equal in an ideal horizontal mass–spring oscillator of amplitude A > 0?

With displacement x measured from equilibrium, U = kx²/2 and total E = kA²/2. Setting U = K = E/2 gives x = ±A/√2. At x = A/2 instead, U = E/4 and K = 3E/4.

What is Gauss’s law, and what field magnitudes follow for ideal spherical, line and sheet charge distributions in vacuum?

∮E·dA = Qenc/ε₀. Isolated conducting sphere in electrostatic equilibrium: E = 0 for r R. Uniform insulating solid sphere: E = k|Q|r/R³ inside. Infinite uniform line: E = |λ|/(2πε₀r). Infinite uniform sheet: E = |σ|/(2ε₀) on each side. Two infinite parallel sheets with opposite densities ±σ give E = |σ|/ε₀ between them. Directions follow charge signs and symmetry.

A point-like clay mass m moving horizontally at v₀ sticks to the bottom of an initially resting vertical uniform rod of mass M and length L. What is their immediate angular speed about the fixed top pivot?

For a brief collision with negligible external angular impulse about the pivot, mv₀L = (ML²/3 + mL²)ω. Thus ω = mv₀/[(M/3 + m)L]. The pivot’s impulse can change total linear momentum, but gives no torque about that pivot. Sticking does not conserve kinetic energy.

How is angular momentum defined, and when is it conserved?

For a particle about a fixed origin, L = r × p. For rotation about a principal axis, the axial component is L = Iω. Zero net external torque about a fixed inertial origin conserves total angular momentum. Constant straight-line momentum can give constant nonzero L about an off-line origin. At fixed nonzero L, reducing I increases angular speed |ω| and rotational energy L²/(2I); internal work can supply that energy.

What are the charging and discharging equations for one ideal series RC circuit?

Let R > 0, C > 0 and τ = RC. Charging from zero through R from constant emf ε: Q = Cε(1 − e^(−t/τ)), VC = Q/C and I = (ε/R)e^(−t/τ). Initially VC = 0; at long times I → 0 and VC → ε. With the source removed, discharge from Q₀ gives Q = Q₀e^(−t/τ) and I = dQ/dt = −[Q₀/(RC)]e^(−t/τ). The negative sign uses the charging-current direction.

How do constant effective gravity changes affect an ideal spring oscillator and a small-angle pendulum?

A fixed-support linear mass–spring oscillator keeps T = 2π√(m/k); constant effective gravity shifts its equilibrium. A pendulum has T ≈ 2π√(L/geff): an elevator accelerating upward at constant a gives geff = g + a and a shorter period. Taking lunar gravity as g/6 gives a period √6 times as long. At geff = 0 there is no gravitational restoring torque for this pendulum model.

How are current, wire resistance, Ohm’s law and resistor power expressed?

I = dQ/dt; for uniform carrier flow, its magnitude is n|q|v_dA, with carrier density n and drift speed v_d. A uniform wire has R = ρL/A. An ohmic resistor has V = IR at fixed physical conditions. Its absorbed power is P = IV = I²R = V²/R. These comparisons use the actual operating resistance; lamp brightness is not a universal fixed-resistance rule.

What static friction and minimum coefficient let a uniform solid cylinder roll down a stationary incline of angle θ without slipping?

For ideal rolling driven only by gravity, with 0 < θ < π/2, a = (2/3)g sin θ. Static friction acts uphill with magnitude fₛ = Mg sin θ/3. Since N = Mg cos θ and fₛ ≤ μₛN, the minimum coefficient is μₛ = (tan θ)/3. Rolling resistance is neglected.

A uniform horizontal beam of length L and mass M is hinged at the left and supported at the right by a cable pulling up and left at angle θ to the horizontal. What are the support forces?

Assume only the beam’s weight, cable tension and hinge force, with 0 < θ ≤ π/2. Torque balance gives (T sin θ)L = MgL/2, so T = Mg/(2 sin θ). The hinge force components are Fx = Mg cot θ/2 to the right and Fy = Mg/2 upward.

What is the electrostatic energy of three point charges separated by distances r₁₂, r₁₃ and r₂₃?

Taking zero energy for infinite separation, U = k(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃). Count each distinct pair once and retain charge signs. This is the external work for quasistatic assembly without losses or a change in kinetic energy; U can be negative.

Why does a uniform solid sphere launched without spin at speed v₀ > 0 on a rough horizontal surface start rolling at speed 5v₀/7?

With constant kinetic coefficient μₖ > 0 and I = 2MR²/5, friction reduces v = v₀ − μₖgt and increases rolling-direction angular speed ω = 5μₖgt/(2R). Setting v = Rω gives t = 2v₀/(7μₖg) and v = 5v₀/7. In this ideal model, with no other horizontal force or rolling resistance, friction is zero after slipping ends. Real bowling balls need not have this inertia.

How are electrostatic potential, potential energy, work and electric field related?

For finite point-charge systems with V = 0 at infinity, V = Σkqᵢ/rᵢ; for a pair, U = kq₁q₂/r. Moving charge q through ΔV changes its energy by qΔV, so field work is −qΔV. E = −∇V, or Eₓ = −dV/dx in one dimension. Wherever E is nonzero, it is perpendicular to equipotential surfaces and points toward decreasing potential.

At the centre of a square with two charges +q and two −q at its corners, when are electric potential and field zero?

With q > 0 and potential zero at infinity, all corners are equally distant, so V = kΣqᵢ/r = 0 for any arrangement. The vector field cancels if identical charges occupy opposite corners. If the positive charges occupy one side and the negatives the opposite side, the field is nonzero and points toward the negative side. Zero V does not imply zero E.

What acceleration does a hanging mass m have when its string unwinds a uniform solid-disk pulley of mass M and radius R?

Assume a fixed frictionless axle, a massless inextensible string and no slip. For fixed-axis rotation, net torque = Iα, with I = MR²/2. Downward force balance gives mg − T = ma; pulley rotation gives TR = I(a/R). Thus a = mg/(m + M/2), downward.

AP Physics 1 & C: Rotational Dynamics, Simple Harmonic Motion, Electrostatics & DC Circuits

32 review cards on rigid-body rotation, harmonic oscillators, electric fields and potentials, capacitors, and DC/RC circuits. Assumes algebra, vectors and basic calculus; emphasizes model assumptions and worked relationships.

Level: intermediate
Languages: en → en
Cards: 32
Markdown: AP Physics 1 & C: Rotational Dynamics, Simple Harmonic Motion, Electrostatics & DC Circuits (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do ideal capacitances combine in parallel and in series?	Parallel: Ceq = ΣCᵢ; voltages are equal and plate charges add. Series: 1/Ceq = Σ(1/Cᵢ); voltage drops add. Equal charge magnitudes in a series chain assume the isolated intermediate conductors have zero net charge, as for capacitors initially uncharged before connection to a battery.
2	How do the formulas for a parallel-plate capacitor distinguish fixed-voltage and fixed-charge changes?	Ignoring edge effects, with a uniform linear dielectric filling the gap: C = κε₀A/d, Q = CV, E = \|V\|/d and U = CV²/2 = Q²/(2C). A connected ideal battery fixes V, so Q and U track C. An isolated capacitor fixes Q, so V and U vary as 1/C. At fixed geometry, replacing vacuum by dielectric κ > 1 multiplies C by κ; E stays fixed with the battery connected and is divided by κ when isolated. At fixed κ and A, increasing d lowers C.
3	How are rotational work, power and kinetic-energy change related for a rigid body rotating about a fixed axis?	Net work is Wnet = ∫τnet dθ = Δ(Iω²/2), with constant I. Only for constant net torque does it reduce to Wnet = τnetΔθ. Instantaneous power is P = τω for torque and angular velocity components along the same fixed axis, with consistent signs.
4	How are moment of inertia and the parallel-axis theorem defined, and how do four standard shapes compare?	I = Σmᵢrᵢ² = ∫r² dm, with r measured perpendicular to the chosen axis. For an axis parallel to a centre-of-mass axis at separation d, I = Icm + Md². For equal M and R, uniform bodies rank: thin hoop MR² > thin spherical shell 2MR²/3 > solid disk/cylinder MR²/2 > solid sphere 2MR²/5. Use the hoop/disk axis through its centre and normal to its plane, the cylinder’s central longitudinal axis, and sphere diameters.
5	What defines simple harmonic motion, and what are the ideal spring and pendulum periods?	SHM obeys d²x/dt² = −ω²x, with T = 2π/ω and f = 1/T. Ideal massless linear spring: T = 2π√(m/k). Small-angle simple pendulum: T ≈ 2π√(L/g). Small-angle physical pendulum: T ≈ 2π√[I/(mgd)], where I is about the pivot and d is the pivot-to-centre-of-mass distance. Linear torsional pendulum: T = 2π√(I/κ), with restoring torque −κθ. Damping is neglected.
6	An ideal source ε feeds R₁, then two parallel branches: R₂ alone and R₃ in series with an initially uncharged capacitor C. What are the initial and final source currents and final capacitor charge?	For positive resistances and an ideal capacitor: at t = 0⁺, VC = 0, so I₀ = ε/[R₁ + R₂R₃/(R₂ + R₃)]. At DC steady state the R₃–C branch carries no current, giving I∞ = ε/(R₁ + R₂). With no drop across R₃, VC = εR₂/(R₁ + R₂), so Qf = CεR₂/(R₁ + R₂).
7	For a point at fixed radius r > 0 rotating about a fixed axis, how do angular and linear quantities relate?	With angles in radians, ω = dθ/dt and α = dω/dt. Signed arc displacement is Δs = rΔθ; tangential velocity is vₜ = rω, while speed is r\|ω\|. Tangential acceleration is aₜ = rα; inward radial acceleration has magnitude aᵣ = rω². The total acceleration magnitude is √(aₜ² + aᵣ²).
8	What are Kirchhoff’s rules, resistor-combination rules and ideal meter connections for ordinary lumped DC circuits?	At a node, total current in equals total current out. Around a loop, signed potential changes sum to zero. Series resistors add: Req = ΣRᵢ; parallel resistors obey 1/Req = Σ(1/Rᵢ). Adding a positive parallel resistance lowers Req and increases current if supply voltage is fixed. An ideal ammeter is connected in series with zero resistance; an ideal voltmeter in parallel with infinite resistance.
9	What are E(r) and V(r) for an isolated conducting sphere of radius R and charge Q > 0 in electrostatic equilibrium?	In vacuum, with no external fields and V(∞) = 0: inside, E = 0 and V = kQ/R; outside (r > R), E = kQ/r² radially outward and V = kQ/r. Potential is continuous at r = R. The field jumps there: its inner limit is zero and its outer limit is kQ/R².
10	What determines motion down an incline for a rigid body with Icm = cMR² rolling without slipping from rest?	On a stationary incline of angle θ, v = Rω and K = Mv²(1 + c)/2. With sufficient static friction and negligible rolling resistance, a = g sin θ/(1 + c); after dropping height h, v = √[2gh/(1 + c)]. Smaller c gives faster descent: solid sphere, disk, thin spherical shell, then hoop. A frictionless sliding block is faster still. Ideal static friction does no work at the stationary contact point.
11	What remains after a capacitor C₁ charged to V₀ is isolated from its battery and connected in parallel to an uncharged C₂?	After transients settle, charge conservation gives Vf = C₁V₀/(C₁ + C₂), Q₁ = C₁Vf and Q₂ = C₂Vf. The final stored energy is Uf = [C₁/(C₁ + C₂)]Ui. The difference becomes heat and/or radiation in a real connection; total energy is conserved. A perfectly lossless model need not settle to this static final state.
12	What are the moments of inertia of a uniform thin rod of mass M and length L about perpendicular axes through its centre and one end?	Through the centre: Icm = ML²/12. Through one end: Iend = Icm + M(L/2)² = ML²/3. Both axes are perpendicular to the rod; the result depends on that orientation.
13	An ideal constant-voltage DC source supplies R₁ in series with R₂. What happens to the power in R₁ and R₂ when a positive resistor R₃ is switched in parallel with R₂?	With fixed positive resistances, the parallel combination lowers total resistance, so current through R₁ rises and P₁ = I²R₁ increases. The larger drop across R₁ leaves less voltage across R₂, so P₂ = V₂²/R₂ decreases. This compares resistor powers; a bulb-brightness interpretation requires an appropriate lamp model.
14	How does a battery’s terminal voltage depend on current in the constant-emf, constant-internal-resistance model?	During discharge, Vₜ = ε − Ir. A Vₜ-versus-I graph has intercept ε and slope −r. This is a model with fixed ε and r; real batteries can change with state of charge, temperature and current.
15	For an ideal undamped mass–spring oscillator, what are x(t), v(t), a(t), the maxima and total energy?	Measured from equilibrium: x = A cos(ωt + φ), v = −Aω sin(ωt + φ), and a = −ω²x, with ω² = k/m. Maximum speed is Aω at x = 0; maximum acceleration magnitude is Aω² at x = ±A. Energy is E = kA²/2 = kx²/2 + mv²/2. Release from rest at x = A sets φ = 0 modulo 2π. Doubling A quadruples E but leaves the ideal period unchanged.
16	What are Coulomb’s law and the electric field of a stationary point charge in vacuum?	The force magnitude is F = k\|q₁q₂\|/r² along the line between the charges: like signs repel and unlike signs attract. Here k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C². Electric field is force per test charge, E = F/q₀ as a vector relation. A source Q gives magnitude k\|Q\|/r², outward for Q > 0 and inward for Q < 0; fields add as vectors.
17	What is torque, and what conditions keep an initially stationary rigid body in static equilibrium?	Torque is the vector cross product τ = r × F. Its magnitude is rF sin φ = Fℓ, where ℓ is the perpendicular lever arm in metres; torque is measured in N·m. Net external force and net external torque must both be zero. With force balance, torque balance holds about any fixed origin.
18	What are the axial field and potential of a uniformly charged thin ring of radius R and total charge Q > 0?	In vacuum, at axial coordinate z with origin at the centre: E_z = kQz/(R² + z²)^(3/2), and V = kQ/√(R² + z²), with V(∞) = 0. Transverse field components cancel. At z = 0, E = 0 but V = kQ/R. Far along the positive axis (z ≫ R), E_z ≈ kQ/z².
19	Where are kinetic and elastic potential energies equal in an ideal horizontal mass–spring oscillator of amplitude A > 0?	With displacement x measured from equilibrium, U = kx²/2 and total E = kA²/2. Setting U = K = E/2 gives x = ±A/√2. At x = A/2 instead, U = E/4 and K = 3E/4.
20	What is Gauss’s law, and what field magnitudes follow for ideal spherical, line and sheet charge distributions in vacuum?	∮E·dA = Qenc/ε₀. Isolated conducting sphere in electrostatic equilibrium: E = 0 for r < R and k\|Q\|/r² for r > R. Uniform insulating solid sphere: E = k\|Q\|r/R³ inside. Infinite uniform line: E = \|λ\|/(2πε₀r). Infinite uniform sheet: E = \|σ\|/(2ε₀) on each side. Two infinite parallel sheets with opposite densities ±σ give E = \|σ\|/ε₀ between them. Directions follow charge signs and symmetry.
21	A point-like clay mass m moving horizontally at v₀ sticks to the bottom of an initially resting vertical uniform rod of mass M and length L. What is their immediate angular speed about the fixed top pivot?	For a brief collision with negligible external angular impulse about the pivot, mv₀L = (ML²/3 + mL²)ω. Thus ω = mv₀/[(M/3 + m)L]. The pivot’s impulse can change total linear momentum, but gives no torque about that pivot. Sticking does not conserve kinetic energy.
22	How is angular momentum defined, and when is it conserved?	For a particle about a fixed origin, L = r × p. For rotation about a principal axis, the axial component is L = Iω. Zero net external torque about a fixed inertial origin conserves total angular momentum. Constant straight-line momentum can give constant nonzero L about an off-line origin. At fixed nonzero L, reducing I increases angular speed \|ω\| and rotational energy L²/(2I); internal work can supply that energy.
23	What are the charging and discharging equations for one ideal series RC circuit?	Let R > 0, C > 0 and τ = RC. Charging from zero through R from constant emf ε: Q = Cε(1 − e^(−t/τ)), VC = Q/C and I = (ε/R)e^(−t/τ). Initially VC = 0; at long times I → 0 and VC → ε. With the source removed, discharge from Q₀ gives Q = Q₀e^(−t/τ) and I = dQ/dt = −[Q₀/(RC)]e^(−t/τ). The negative sign uses the charging-current direction.
24	How do constant effective gravity changes affect an ideal spring oscillator and a small-angle pendulum?	A fixed-support linear mass–spring oscillator keeps T = 2π√(m/k); constant effective gravity shifts its equilibrium. A pendulum has T ≈ 2π√(L/geff): an elevator accelerating upward at constant a gives geff = g + a and a shorter period. Taking lunar gravity as g/6 gives a period √6 times as long. At geff = 0 there is no gravitational restoring torque for this pendulum model.
25	How are current, wire resistance, Ohm’s law and resistor power expressed?	I = dQ/dt; for uniform carrier flow, its magnitude is n\|q\|v_dA, with carrier density n and drift speed v_d. A uniform wire has R = ρL/A. An ohmic resistor has V = IR at fixed physical conditions. Its absorbed power is P = IV = I²R = V²/R. These comparisons use the actual operating resistance; lamp brightness is not a universal fixed-resistance rule.
26	What static friction and minimum coefficient let a uniform solid cylinder roll down a stationary incline of angle θ without slipping?	For ideal rolling driven only by gravity, with 0 < θ < π/2, a = (2/3)g sin θ. Static friction acts uphill with magnitude fₛ = Mg sin θ/3. Since N = Mg cos θ and fₛ ≤ μₛN, the minimum coefficient is μₛ = (tan θ)/3. Rolling resistance is neglected.
27	A uniform horizontal beam of length L and mass M is hinged at the left and supported at the right by a cable pulling up and left at angle θ to the horizontal. What are the support forces?	Assume only the beam’s weight, cable tension and hinge force, with 0 < θ ≤ π/2. Torque balance gives (T sin θ)L = MgL/2, so T = Mg/(2 sin θ). The hinge force components are Fx = Mg cot θ/2 to the right and Fy = Mg/2 upward.
28	What is the electrostatic energy of three point charges separated by distances r₁₂, r₁₃ and r₂₃?	Taking zero energy for infinite separation, U = k(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃). Count each distinct pair once and retain charge signs. This is the external work for quasistatic assembly without losses or a change in kinetic energy; U can be negative.
29	Why does a uniform solid sphere launched without spin at speed v₀ > 0 on a rough horizontal surface start rolling at speed 5v₀/7?	With constant kinetic coefficient μₖ > 0 and I = 2MR²/5, friction reduces v = v₀ − μₖgt and increases rolling-direction angular speed ω = 5μₖgt/(2R). Setting v = Rω gives t = 2v₀/(7μₖg) and v = 5v₀/7. In this ideal model, with no other horizontal force or rolling resistance, friction is zero after slipping ends. Real bowling balls need not have this inertia.
30	How are electrostatic potential, potential energy, work and electric field related?	For finite point-charge systems with V = 0 at infinity, V = Σkqᵢ/rᵢ; for a pair, U = kq₁q₂/r. Moving charge q through ΔV changes its energy by qΔV, so field work is −qΔV. E = −∇V, or Eₓ = −dV/dx in one dimension. Wherever E is nonzero, it is perpendicular to equipotential surfaces and points toward decreasing potential.
31	At the centre of a square with two charges +q and two −q at its corners, when are electric potential and field zero?	With q > 0 and potential zero at infinity, all corners are equally distant, so V = kΣqᵢ/r = 0 for any arrangement. The vector field cancels if identical charges occupy opposite corners. If the positive charges occupy one side and the negatives the opposite side, the field is nonzero and points toward the negative side. Zero V does not imply zero E.
32	What acceleration does a hanging mass m have when its string unwinds a uniform solid-disk pulley of mass M and radius R?	Assume a fixed frictionless axle, a massless inextensible string and no slip. For fixed-axis rotation, net torque = Iα, with I = MR²/2. Downward force balance gives mg − T = ma; pulley rotation gives TR = I(a/R). Thus a = mg/(m + M/2), downward.