What are Newton's gravitational force law and the field of a spherical mass distribution?

For point masses, attractive force magnitude F = GMm/r². Outside a spherically symmetric source, g = GM/r² inward. Inside, g = GM_enc(r)/r² for r > 0, using the mass enclosed within r.

What is the work–energy theorem for a constant-mass Newtonian particle?

Net work equals the change in kinetic energy: W_net = ΔK = ½m(v_f² − v_i²).

How does net impulse relate to momentum?

J = ∫F_net dt = F_avgΔt = Δp. For constant mass, Δp = m(v_f − v_i). These are vector relations.

How do scalar and vector quantities differ?

A scalar has a numerical value and unit; a vector also has direction. Speed is scalar, while velocity is a vector.

For a fixed Newtonian system of constant-mass particles, how are center-of-mass position and acceleration found?

R_cm = Σmᵢrᵢ/M, with M = Σmᵢ. If internal forces cancel in pairs, M a_cm = ΣF_ext.

Two particles stick together in a collision with negligible external impulse. What is their common final velocity?

v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂), a vector relation. This is a perfectly inelastic collision; translational kinetic energy is generally reduced.

What is Hooke's law for an ideal linear spring, with x measured from its natural length?

Fₛ = −kx, where k is stiffness. The minus sign means the spring force opposes extension or compression, toward the natural length.

How are instantaneous velocity and acceleration defined using calculus?

v = dr/dt and a = dv/dt = d²r/dt², where r is the position vector and the derivatives exist.

How do centripetal and centrifugal force differ?

Centripetal force is the net inward force component causing circular acceleration in an inertial frame. Centrifugal force is an outward inertial term introduced in a rotating frame, not a new physical interaction.

Which gravitational potential-energy formulas apply near Earth's surface and to point masses?

In a uniform near-surface field, U = mgh relative to h = 0. For point masses, U = −GMm/r with zero at infinite separation. It also applies to nonoverlapping spherically symmetric mass distributions, with r their center-to-center separation.

How do elastic and inelastic collisions differ when external impulse is negligible?

Total momentum is conserved in both. An elastic collision also conserves total kinetic energy; an inelastic collision does not. Ordinary dissipative collisions convert some kinetic energy into internal energy.

For projectile motion without air resistance and with uniform downward g, what are the components?

Taking upward y as positive: aₓ = 0, vₓ = v₀cosθ, aᵧ = −g and vᵧ = v₀sinθ − gt. θ is the launch angle above horizontal.

With a time-independent potential in the usual particle model, when is mechanical energy K + U conserved?

If only conservative forces do work, K + U is constant. More generally, Δ(K + U) = W_nc, so zero total nonconservative work is sufficient.

For a person on an elevator scale, with upward acceleration aᵧ positive, what is the normal force N?

While contact is maintained, N = m(g + aᵧ). Upward acceleration increases N; downward acceleration reduces it. Constant velocity gives N = mg; free fall has aᵧ = −g and N = 0.

What elastic potential energy is stored in an ideal linear spring?

U = ½kx², taking zero at the natural length. x is extension or compression from that length; k is stiffness.

For fixed-radius circular motion, how are angular rates related to tangential motion and total acceleration magnitude?

Signed tangential components satisfy v_t = rω and a_t = rα. With speed v = |v_t|, inward a_r = v²/r. The total magnitude is |a| = √(a_t² + a_r²), since the components are perpendicular.

What is Newton's version of Kepler's third law for an isolated circular two-body orbit?

T² = 4π²r³/[G(M + m)], where r is the constant separation and T the period. If m ≪ M, the denominator is approximately GM.

What is linear momentum, and when is a fixed system's total momentum conserved?

p = mv is a vector. For the system, dP/dt = ΣF_ext; P is constant when net external force is zero. Zero net external impulse ensures equal initial and final P.

What equations describe one-dimensional motion with constant acceleration a?

For constant acceleration: v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx; Δx = ½(v₀ + v)t. Here t is elapsed time and v₀ is initial velocity.

What is Tsiolkovsky's equation for an ideal straight rocket burn, and what assumptions apply?

Δv = v_e ln(m₀/m_f), with constant effective exhaust speed v_e relative to the rocket and negligible external forces. m₀ and m_f are total rocket masses; m_f can include remaining propellant and need not equal dry mass.

For fixed launch speed, equal launch and landing heights, uniform g and no drag, which angle maximizes range?

R = v₀²sin(2θ)/g, so θ = 45° maximizes range. Complementary angles θ and 90° − θ have equal ranges under these assumptions.

How is the instantaneous mechanical power of a force calculated?

P = dW/dt = F·v = |F||v|cosθ. F·v is a vector dot product; θ is the angle between force and velocity. 1 W = 1 J/s.

In an ideal Atwood machine with m₁ > m₂, a taut massless inextensible string, no friction and negligible pulley inertia, what are a and T?

a = (m₁ − m₂)g/(m₁ + m₂), with the heavier mass accelerating downward. T = 2m₁m₂g/(m₁ + m₂).

How does the work of conservative and nonconservative forces differ?

For a time-independent conservative force, work depends only on endpoints and equals −ΔU; closed-path work is zero. Nonconservative work can add or remove mechanical energy.

From the surface of an isolated spherically symmetric body of mass M and radius R, what escape speed applies to a negligible test mass moving under gravity alone?

v_esc = √(2GM/R), measured relative to the source's center. This is the minimum speed for reaching infinity with zero remaining speed in this model.

How do distance, displacement, average speed and average velocity differ?

Distance is total path length; displacement is the vector change in position. Average speed is distance/elapsed time; average velocity is displacement/elapsed time.

Why is uniform circular motion accelerated, and what is its acceleration?

Speed is constant but velocity direction changes. The acceleration points inward, with magnitude a_c = v²/r = ω²r.

Which forces provide inward acceleration for a nonskidding car on a flat curve, a circular-orbit satellite and a conical pendulum?

Car: static friction. Satellite: gravity. Conical pendulum: the horizontal tension component T sinθ, with θ measured from vertical. Centripetal force is not an additional interaction.

How do inertial and passive gravitational mass differ, and what does weak equivalence predict?

Inertial mass relates force to acceleration; passive gravitational mass sets force in a given gravitational field. Weak equivalence predicts identical free fall for test bodies in the same field with the same initial conditions, neglecting nongravitational forces.

What do slopes and signed areas of one-dimensional motion graphs give?

x–t slope gives velocity; v–t slope gives acceleration. Signed area under v–t gives displacement; signed area under a–t gives change in velocity.

When can a block stay at rest on a fixed incline at angle θ to horizontal, with only gravity and contact forces?

N = mg cosθ and static friction acts uphill with fₛ = mg sinθ, provided fₛ ≤ μₛN. Otherwise rest is impossible. In the simple sliding model, fₖ = μₖN opposes relative slipping.

What force balance defines terminal speed for an object falling through a still fluid?

The velocity is steady and net force is zero: mg = buoyancy + drag. If buoyancy is negligible, drag = mg. The drag law depends on the flow regime.

Classical Mechanics: Kinematics, Dynamics, Energy & Momentum

32 review cards for upper-secondary or introductory college mechanics: motion, forces, energy, momentum, circular motion and gravity. Some cards use calculus.

Level: basic
Languages: en → en
Cards: 32
Markdown: Classical Mechanics: Kinematics, Dynamics, Energy & Momentum (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	What are Newton's gravitational force law and the field of a spherical mass distribution?	For point masses, attractive force magnitude F = GMm/r². Outside a spherically symmetric source, g = GM/r² inward. Inside, g = GM_enc(r)/r² for r > 0, using the mass enclosed within r.
2	What is the work–energy theorem for a constant-mass Newtonian particle?	Net work equals the change in kinetic energy: W_net = ΔK = ½m(v_f² − v_i²).
3	How does net impulse relate to momentum?	J = ∫F_net dt = F_avgΔt = Δp. For constant mass, Δp = m(v_f − v_i). These are vector relations.
4	How do scalar and vector quantities differ?	A scalar has a numerical value and unit; a vector also has direction. Speed is scalar, while velocity is a vector.
5	For a fixed Newtonian system of constant-mass particles, how are center-of-mass position and acceleration found?	R_cm = Σmᵢrᵢ/M, with M = Σmᵢ. If internal forces cancel in pairs, M a_cm = ΣF_ext.
6	Two particles stick together in a collision with negligible external impulse. What is their common final velocity?	v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂), a vector relation. This is a perfectly inelastic collision; translational kinetic energy is generally reduced.
7	What is Hooke's law for an ideal linear spring, with x measured from its natural length?	Fₛ = −kx, where k is stiffness. The minus sign means the spring force opposes extension or compression, toward the natural length.
8	How are instantaneous velocity and acceleration defined using calculus?	v = dr/dt and a = dv/dt = d²r/dt², where r is the position vector and the derivatives exist.
9	How do centripetal and centrifugal force differ?	Centripetal force is the net inward force component causing circular acceleration in an inertial frame. Centrifugal force is an outward inertial term introduced in a rotating frame, not a new physical interaction.
10	Which gravitational potential-energy formulas apply near Earth's surface and to point masses?	In a uniform near-surface field, U = mgh relative to h = 0. For point masses, U = −GMm/r with zero at infinite separation. It also applies to nonoverlapping spherically symmetric mass distributions, with r their center-to-center separation.
11	How do elastic and inelastic collisions differ when external impulse is negligible?	Total momentum is conserved in both. An elastic collision also conserves total kinetic energy; an inelastic collision does not. Ordinary dissipative collisions convert some kinetic energy into internal energy.
12	For projectile motion without air resistance and with uniform downward g, what are the components?	Taking upward y as positive: aₓ = 0, vₓ = v₀cosθ, aᵧ = −g and vᵧ = v₀sinθ − gt. θ is the launch angle above horizontal.
13	With a time-independent potential in the usual particle model, when is mechanical energy K + U conserved?	If only conservative forces do work, K + U is constant. More generally, Δ(K + U) = W_nc, so zero total nonconservative work is sufficient.
14	For a person on an elevator scale, with upward acceleration aᵧ positive, what is the normal force N?	While contact is maintained, N = m(g + aᵧ). Upward acceleration increases N; downward acceleration reduces it. Constant velocity gives N = mg; free fall has aᵧ = −g and N = 0.
15	What elastic potential energy is stored in an ideal linear spring?	U = ½kx², taking zero at the natural length. x is extension or compression from that length; k is stiffness.
16	For fixed-radius circular motion, how are angular rates related to tangential motion and total acceleration magnitude?	Signed tangential components satisfy v_t = rω and a_t = rα. With speed v = \|v_t\|, inward a_r = v²/r. The total magnitude is \|a\| = √(a_t² + a_r²), since the components are perpendicular.
17	What is Newton's version of Kepler's third law for an isolated circular two-body orbit?	T² = 4π²r³/[G(M + m)], where r is the constant separation and T the period. If m ≪ M, the denominator is approximately GM.
18	What is linear momentum, and when is a fixed system's total momentum conserved?	p = mv is a vector. For the system, dP/dt = ΣF_ext; P is constant when net external force is zero. Zero net external impulse ensures equal initial and final P.
19	What equations describe one-dimensional motion with constant acceleration a?	For constant acceleration: v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx; Δx = ½(v₀ + v)t. Here t is elapsed time and v₀ is initial velocity.
20	What is Tsiolkovsky's equation for an ideal straight rocket burn, and what assumptions apply?	Δv = v_e ln(m₀/m_f), with constant effective exhaust speed v_e relative to the rocket and negligible external forces. m₀ and m_f are total rocket masses; m_f can include remaining propellant and need not equal dry mass.
21	For fixed launch speed, equal launch and landing heights, uniform g and no drag, which angle maximizes range?	R = v₀²sin(2θ)/g, so θ = 45° maximizes range. Complementary angles θ and 90° − θ have equal ranges under these assumptions.
22	How is the instantaneous mechanical power of a force calculated?	P = dW/dt = F·v = \|F\|\|v\|cosθ. F·v is a vector dot product; θ is the angle between force and velocity. 1 W = 1 J/s.
23	In an ideal Atwood machine with m₁ > m₂, a taut massless inextensible string, no friction and negligible pulley inertia, what are a and T?	a = (m₁ − m₂)g/(m₁ + m₂), with the heavier mass accelerating downward. T = 2m₁m₂g/(m₁ + m₂).
24	How does the work of conservative and nonconservative forces differ?	For a time-independent conservative force, work depends only on endpoints and equals −ΔU; closed-path work is zero. Nonconservative work can add or remove mechanical energy.
25	From the surface of an isolated spherically symmetric body of mass M and radius R, what escape speed applies to a negligible test mass moving under gravity alone?	v_esc = √(2GM/R), measured relative to the source's center. This is the minimum speed for reaching infinity with zero remaining speed in this model.
26	How do distance, displacement, average speed and average velocity differ?	Distance is total path length; displacement is the vector change in position. Average speed is distance/elapsed time; average velocity is displacement/elapsed time.
27	Why is uniform circular motion accelerated, and what is its acceleration?	Speed is constant but velocity direction changes. The acceleration points inward, with magnitude a_c = v²/r = ω²r.
28	Which forces provide inward acceleration for a nonskidding car on a flat curve, a circular-orbit satellite and a conical pendulum?	Car: static friction. Satellite: gravity. Conical pendulum: the horizontal tension component T sinθ, with θ measured from vertical. Centripetal force is not an additional interaction.
29	How do inertial and passive gravitational mass differ, and what does weak equivalence predict?	Inertial mass relates force to acceleration; passive gravitational mass sets force in a given gravitational field. Weak equivalence predicts identical free fall for test bodies in the same field with the same initial conditions, neglecting nongravitational forces.
30	What do slopes and signed areas of one-dimensional motion graphs give?	x–t slope gives velocity; v–t slope gives acceleration. Signed area under v–t gives displacement; signed area under a–t gives change in velocity.
31	When can a block stay at rest on a fixed incline at angle θ to horizontal, with only gravity and contact forces?	N = mg cosθ and static friction acts uphill with fₛ = mg sinθ, provided fₛ ≤ μₛN. Otherwise rest is impossible. In the simple sliding model, fₖ = μₖN opposes relative slipping.
32	What force balance defines terminal speed for an object falling through a still fluid?	The velocity is steady and net force is zero: mg = buoyancy + drag. If buoyancy is negligible, drag = mg. The drag law depends on the flow regime.