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Classical Mechanics II Flashcards

Apply advanced mechanics principles to analyze motion in central forces, non-inertial frames, fluids, and 3D systems using Newtonian and analytical formalisms. (65 cards)

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1
Q

Define:

central force

A

A force that always points towards or away from a fixed point and whose magnitude depends only on the distance from that point.

Central forces are conservative - their work done depends only on the radial coordinate and therefore is path independent.

They also generate motion confined to a plane with conserved angular momentum - this follows by looking at the time-derivative of the angular momentum and using Newton’s second law.

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2
Q

True or False:

For an object in a closed orbit under a central gravitational force, the total mechanical energy is positive.

A

False

In a closed (elliptical or circular) orbit under gravity, the total mechanical energy is negative. This indicates a bound system, where the object lacks enough energy to escape the gravitational potential well.

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3
Q

Find the angular momentum L of a particle of mass m at the perihelion in an orbit governed by a central force.

Assume distance at perihelion (rₚ) and velociy (vₚ) are known.

A

Perihelion is the distance of closest approach to the Sun. Taking the origin of the coordinate system to be at the center of Sun (assuming the mass of the particle to be negligible compared to the mass of the Sun), the position vector of the particle is perpendicular to its velocity at the perihelion.

Using the definition of the angular momentum, we then get the required result. Note that this angular momentum is conserved as the particle moves in its orbit.

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4
Q

What is the ratio T1/T 2 of the orbital periods for two planets orbiting the same star?

Assume r1=4a and r2=a.

A

By Kepler’s Third Law: T² ∝ r³, so (T1/T2)2 = (4a/a)³ = 64 ⇒ T1/T2 = 8.

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5
Q

True or False:

A planet in a circular orbit with a smaller radius will take longer to complete an orbit than a planet with a larger radius.

A

False

According to Kepler’s third law, smaller orbital radii lead to shorter periods.

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6
Q

Fill in the blank:

The central force associated with the potential U(r)=-k/r is given by ______?

A

Central forces can be derived as the negative gradient of the potential energy function. Since the potential only depends on the radial coordinate, the derivative with respect to this coordinate needs to be taken.

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7
Q

Two identical satellites orbit Earth in circular orbits. B’s orbital radius is twice A’s. What is the ratio LA/LB of their angular momenta?

A

  • For circular orbits, the centripetal force must be equal to the gravitational force.
  • This gives us that v = √(GM/r).
  • Then, the angular momentum L = mrv = mr√(GM/r)
  • Therefore, L ∝ √r.

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8
Q

What shape do bound orbits take under an inverse-square central force such as gravity?

A

Ellipses

Newton showed that the inverse-square law implies Kepler’s laws. Kepler’s first law says that the orbits are ellipses.

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9
Q

For a particle at the surface of the Earth, what is the escape velocity?

Assume the radius of the Earth is R, and it’s mass is M.

A

The total energy of the particle is E= 1/2 mv² - (GMm)/R where m is the mass of the particle.

To just escape the influence of the Earth, the energy of the particle infinitely far away from the Earth will be zero (the particle will be barely moving, and we have set the potential energy to be zero at infinity).

Then using conservation of energy, we find the desired result.

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10
Q

A particle moves under of a force with the corresponding potential energy U(r)= 1/r² - 1/r.

What is the magnitude of the maximum attractive force?

A

Although the force is negative (pointing inward) and indicates attraction, the magnitude is positive.

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11
Q

What does the effective potential in central force motion represent?

A

The combination of actual potential energy and centrifugal potential energy.

The equation Ueff(r)=U(r)+L²/(2μr²) is used to reduce orbital motion to a 1D problem.

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12
Q

Fill in the blank:

Circular orbits under an attractive 1/r potential are stable because small radial displacements result in ______ forces.

A

restoring

A circular orbit is stable if a small radial displacement (inward or outward) produces a force that pushes the particle back toward the equilibrium radius, which is a restoring force.

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13
Q

What does Foucault’s pendulum demonstrate about Earth’s motion?

A

It provides direct evidence of Earth’s rotation.

The precession of the swing plane reflects the Coriolis effect due to Earth’s rotation.

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14
Q

A Foucault pendulum is set up at latitude θ. What is the angular rate at which the pendulum’s oscillation plane rotates due to Earth’s rotation?

A

where ω is Earth’s angular velocity

The Coriolis force deflects the pendulum’s plane in the rotating Earth frame, revealing Earth’s rotation through precession dependent on latitude.

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15
Q

True or False:

In all central force problems, total mechanical energy is conserved.

A

True

(for conservative central forces)

Conservative forces include gravitational and electrostatic forces.

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16
Q

What is the total energy of a particle in a circular orbit of radius r (under gravitational attraction) around a planet with mass M?

Assume that the mass of the particle is negligible compared to the mass of the planet.

A

The centripetal force equals the gravitational force.

  • This allows us to find v in terms of G, M, and r.
  • Use this to find the kinetic energy K = GMm/(2r).
  • Also, the potential energy U = -GMm/r.
  • The total energy E = K + U then follows.

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17
Q

What is the Lagrangian of a system in classical mechanics?

A

L = T - V

  • T is kinetic energy
  • V is potential energy

Once we know the Lagrangian, we can derive the equations of motion via the Euler–Lagrange equations. The Lagrangian approach is an alternative to applying Newton’s laws. It is especially useful when there are constraints in our system.

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18
Q

Derive the Lagrangian of a simple pendulum of mass m and length l, under gravity.

A

  • Use polar coordinates: T = (1/2)ml²(dθ/dt)².
  • Setting the potential energy to be zero at the lowest point in the motion, the height the bob goes up is l(1 - cos θ).
  • Remember that the angle is being measured with respect to the vertical. Note that any constant can be added to a Lagrangian without changing the physics.
  • We can also write L = (1/2)ml²(dθ/dt)² + mgl cos θ.

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19
Q

A block slides without friction on an inclined plane. Using x along the slope as the generalized coordinate, find its Lagrangian.

A

Choose axis aligned with motion; the system has one degree of freedom. Note that the height of the block (with our chosen coordinate system) is simply x sin θ.

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20
Q

True or False:

In a system with a cyclic coordinate, the corresponding conjugate momentum is conserved.

A

True

A cyclic coordinate is one that does not appear in the Lagrangian, leading to the conservation of its conjugate momentum. Changing the cyclic coordinate does not change the Lagrangian, which means that we have a symmetry.

This link between symmetry and conserved quantities is elucidated by Noether’s theorem.

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21
Q

A bead of mass m slides on a hoop rotating at constant angular velocity ω.

What is the Lagrangian in terms of the angle θ that the bead makes with the vertical?

A

The kinetic energy has two contributions.

  • The motion along the hoop - this is simple motion with fixed radius, leading to the first term in the Lagrangian.
  • The rotation of the hoop itself. The bead rotates in a horizontal circle of radius R sin θ.

The tangential speed is then ωR sin θ. This leads to the second term in the Lagrangian. The last term is obviously from the gravitational potential energy.

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22
Q

What is the Hamiltonian of a system?

A

Here L is the Lagrangian. Given the generalized coordinates q_i, we get the momenta p_i as the partial derivative of L with respect to the time derivative of the corresponding generalized coordinate. Note that the Hamiltonian is expressed in terms of q i and p i. The Hamiltonian often represents the total energy.

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23
Q

True or False:

The Hamiltonian is always equal to the total energy of the system.

A

False

The Hamiltonian often equals the total energy. However, there are exceptions.

For example, if the Lagrangian depends explicitly on time, the Hamiltonian need not be equal to the total energy. For a charged particle in an electromagnetic field, the Hamiltonian cannot be simply written as (p2)/2m + qϕ, where ϕ is the scalar potential.

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24
Q

Fill in the blank:

The transformation of coordinates that leads to a diagonalized Hamiltonian is called a ______ transformation.

A

canonical

Canonical transformations simplify the form of Hamiltonians, facilitating the solution of complex dynamical systems by reducing them to simpler, often integrable forms.

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25
Find the Hamiltonian for a **1D harmonic oscillator** of mass m and spring constant k.
## Footnote * Start from L = (1/2)m ẋ² − (1/2)kx² * Then p = m ẋ * Apply H = pẋ − L
26
Write the **Lagrangian and generalized momenta** for a free particle in 2D polar coordinates.
## Footnote No potential energy here, only kinetic energy. θ is cyclic, meaning that p θ is constant. This is conservation of angular momentum in fancy form.
27
Which transformation connects the **Lagrangian and Hamiltonian formalisms**?
The **Legendre** transformation. ## Footnote The Legendre transformation converts the function from variables (q, q̇) to (q, p). Legendre transformations are also widely used in thermodynamics.
28
# True or False: The number of Hamilton's equations is twice the number of **degrees of freedom**.
True ## Footnote For n degrees of freedom, there are 2n first-order equations: one for qi, one for pi. Note that there is no conflict with the Euler-Lagrange equations - while there are n of these, these are second order differential equations. Hamilton's equations are first order differential equations.
29
Use a single coordinate x to derive the Lagrangian for an ideal **Atwood machine** with unequal masses m1 and m2.
## Footnote * Let x be the distance that m1 is below the pulley. * Then, placing our coordinate system at the pulley, its coordinate is -x. * The second mass has then coordinate -(l - πR - x), where l is the length of the string, assumed to be constant, and R is the radius of the pulley. ; the other follows by constraint. * Both masses have the same speed since the length of the string is not changing. * We can now write down the Lagrangian, and drop any constant terms.
30
Consider the double pendulum shown in the figure. What would be the **Lagrangian?**
## Footnote This is the classic double pendulum problem. The Lagrangian is the sum of the kinetic energies of m1 and m2 minus the potential energy.
31
If L has **no explicit time** dependence, what quantity is conserved?
Total energy ## Footnote Noether's theorem implies energy conservation for time-invariant Lagrangians.
32
What makes the **three-body problem** fundamentally different from the two-body problem?
It is generally not solvable in closed form due to the **nonlinearity and sensitivity to initial conditions.** ## Footnote Unlike the two-body problem, the three-body problem exhibits chaotic behavior and lacks a general analytical solution. The two-body problem can be solved by mapping to an effective one-body problem via the center of mass and the reduced mass. No such general transformation exists for the three-body problem.
33
A small object is placed between the Earth and the Moon along the line joining their centers. At what type of point does it experience **no net force** in the rotating frame?
At a **Lagrange point**. ## Footnote Lagrange points are equilibrium positions where gravitational and centrifugal forces balance in a rotating frame.
34
# True or False: In the restricted three-body problem, the **total mechanical energy** of the system is conserved.
False ## Footnote In the rotating frame of the restricted problem, the mechanical energy not is conserved. When analyzing the problem from the rotating frame, fictitious forces appear that are non-conservative.
35
What **assumption** simplifies the restricted three-body problem?
One body has **negligible mass** and does not affect the motion of the other two. ## Footnote This allows analysis of a small body (like a satellite) moving under the gravitational field of two large masses.
36
In the circular restricted three-body problem, what **quantity remains constant** and combines kinetic and potential terms?
The Jacobi constant (Cⱼ) ## Footnote Cⱼ= 2Ω - v², where Ω is the effective potential (gravitational plus centrifugal) and v is the velocity in the rotating frame.
37
Two particles of mass m1 and m2 interact via a central force. How does using the **reduced mass simplify the analysis of their 3D motion**?
It reduces the two-body system to a **one-body problem**. ## Footnote * μ=(m1m2)/(m1+m2) * r is the relative position The reduced mass captures the inertia of the relative motion, allowing standard single-particle methods to be used.
38
What does a **non-inertial** reference frame describe?
A frame undergoing acceleration relative to an inertial frame, where **fictitious forces must be introduced**. ## Footnote Forces are interactions, and Newton's second law then says that accelerations are due to interactions. However, in an accelerating frame, we can measure the acceleration of an object to be non-zero even though there is no apparent interaction of this object with anything else. In such frames, Newton’s laws require the inclusion of apparent forces (e.g., Coriolis, centrifugal). For example, consider a book placed in a car moving at constant velocity - the car seat is very smooth, meaning that any friction between the book and the car seat is negligible. If now the brakes are applied, the book is observed to move forward in the reference frame of the car, even though there could have been no horizontal force.
39
Explain the **difference** between centrifugal force and centripetal force in the context of circular motion.
* **Centrifugal force** is a fictitious force perceived in a rotating reference frame, acting radially outward. * **Centripetal force** is the net force acting towards the center of the circular arc an object is moving in. It is required to change the direction of the velocity vector. ## Footnote Centrifugal force is an apparent effect due to the observer's non-inertial frame, while centripetal force involves real interactions, such as tension, gravity, or friction.
40
A ball is dropped from height h at latitude θ. What is the **eastward Coriolis deflection**?
## Footnote This is derived assuming small deflections, constant gravity, and Earth's angular speed ω.
41
In a rotating frame with angular velocity ω, what is the expression for the **centrifugal force** on a particle (of mass m) with position vector **r** in the rotating frame?
## Footnote The centrifugal force appears due to rotation and points outward from the axis.
42
# Fill in the blank: In a **rotating frame**, the total fictitious force includes the Coriolis, centrifugal, and \_\_\_\_\_ forces.
Euler ## Footnote The **Euler force** arises from a non-constant angular velocity of the rotating frame: FE = -m (dω/dt) × r.
43
A projectile is fired northward with velocity v₀ at latitude θ. What is the **Coriolis acceleration**?
## Footnote The Coriolis force acts perpendicular to velocity and rotation axis, deflecting paths in rotating frames.
44
How would you find the **acceleration of a particle** of mass m in a frame rotating with angular velocity ω?
Use **Newton's second law**, but include fictitious (inertial) forces such as the centrifugal and Coriolis forces to account for the rotation. ## Footnote Each fictitious force must be added to apply Newton’s laws in non-inertial rotating frames.
45
Explain the **origin and expression** of the Coriolis force acting on a particle in a rotating reference frame.
It arises due to **motion observed in a rotating frame**. ## Footnote * ω is the angular velocity of the frame * v is the velocity relative to the rotating frame The Coriolis force is a fictitious force that appears in non-inertial frames and deflects motion perpendicular to both velocity and axis of rotation.
46
# True or False: The **Coriolis force** always acts in the direction of motion.
False ## Footnote It acts perpendicular to the velocity and the rotation axis; its direction depends on the **velocity vector** (relative to the rotating frame) and the angular velocity of the frame.
47
A particle moves in a circle of radius R on a horizontal turntable rotating at angular velocity ω. What **fictitious forces** act?
* Centrifugal: mω²R * Coriolis: 2mωvᵣ * Euler: mαR ## Footnote * The centrifugal force acts radially outward. * The Coriolis force is perpendicular to both vr and the axis of rotation, and vᵣ is the velocity relative to the turntable. * The Euler force is tangential, and α is angular acceleration.
48
Compare the Lagrangian of a free particle in an **inertial frame and in a rotating frame.**
## Footnote The square in the Lagrangian for the free particle in a rotating frame expands to include terms from **Coriolis and centrifugal effects**. It includes kinetic contributions from the rotation, leading to fictitious-force-derived terms.
49
Consider two equal masses (with mass m) and three identical springs as shown below. What is the **Hamiltonian** for this system?
## Footnote In this case, the Hamiltonian is simply the total energy.
50
Consider two equal masses (with mass m) and three identical springs as shown below. What are the normal mode frequencies?
## Footnote * Obtain the equations of motion for the masses by using Newton's laws (or use the Lagrangian and then the Euler-Lagrange equations, or use the Hamiltonian and then Hamilton's equations). * Then add the two equations to get the normal modes. * Read off the angular frequencies. * In general, finding the normal modes involves diagonalizing a matrix.
51
What assumptions define an **ideal fluid** in elementary dynamics?
* Incompressible * Non-viscous * Irrotational * Steady ## Footnote These assumptions simplify the Navier-Stokes equations to Euler's equation for ideal flow.
52
Water flows through a horizontal pipe narrowing from radius R to R/2. If the speed in the wide section is v, what is the **pressure difference** between the two sections?
## Footnote * First, use the continuity equation: A1v 1 = A2 v2. * The cross-sectional area of the pipe reduces by a factor of 4, so the velocity increases four times. * Now use Bernoulli's equation. * We have P1 + 0.5 \rho v2 = P2 + 0.5 \rho (4v)2. * Simplify for P1 - P2.
53
# True or False: **Bernoulli’s equation** can be applied to all fluid flows.
False ## Footnote Bernoulli fails for viscous, compressible, unsteady, or non-streamline flows.
54
A tank with height h has a small hole at the bottom. What is the **exit speed of the fluid**?
## Footnote Derived from **Bernoulli’s equation** assuming atmospheric pressure at both top and exit, and negligible velocity for the fluid at the top, then P1 and P2 cancel each other. This is known as Torricelli's law, a special case of Bernoulli's equation.
55
A sphere moves slowly through a fluid with **dynamic viscosity** η. What is the expression for **drag force** at low Reynolds number?
F = 6πηRv ## Footnote This is Stokes’ Law, valid for low Reynolds number.
56
# Fill in the blank: The **continuity equation** expresses conservation of \_\_\_\_\_ in fluid dynamics.
mass ## Footnote ∇⋅**v** = 0 for incompressible fluids. In general, however, the continuity equation is ∂ρ/∂t + ∇ · (ρv) = 0. By doing a volume integral and applying the divergence theorem, we can see that the first term describes the change in the mass of the fluid within the volume, while the second describes the flow out (or in) of the volume. For constant density, we get the result for incompressible fluids.
57
How do we arrive at **Euler's equation** in fluid flow?
Net force on a fluid element **equals the rate of change** of its momentum. ## Footnote This leads to ρD**v**/Dt = -∇P + ρ**g**. Note that the left hand side contains a total derivative. We assume inviscid (that is, non-viscous) flow.
58
A small sphere of radius R falls in a viscous fluid. Derive its **terminal velocity.**
## Footnote At terminal velocity, gravitational force equals viscous drag plus buoyancy. ρs is the density of the sphere, while ρf is the density of the fluid. The buoyant force is given by Archimedes' principle, while the drag force follows from Stokes' law.
59
In a Venturi tube with cross-sections A₁ and A₂, what is the **fluid speed** at A₂ given pressure difference ΔP?
## Footnote Use Bernoulli’s principle combined with A₁v₁ = A₂v₂.
60
Find the **pressure** at a depth h in a fluid of density ρ, open to the atmosphere.
P = P₀ + ρ gh ## Footnote **Hydrostatic pressure** increases linearly with depth.
61
A pipe has cross-section A and fluid speed v. What is the **volumetric flow rate** Q?
Q = Av ## Footnote In general, we take the dot product of the area vector of the surface and the velocity vector.
62
A **capillary tube** of radius r is inserted into water. What is the height the liquid rises, assuming surface tension γ?
## Footnote * Balance the surface tension force and the weight of the column of water. * θ is the contact angle between the water and the tube. * ρ is the density of water. * g is the usual acceleration of free fall. * The upward force is 2πrγcosθ, while the weight is ρπr2hg. * Equate these two forces.
63
What does the **Reynolds number** describe in fluid flow?
The **ratio of inertial** to viscous forces. ## Footnote It indicates flow regime: laminar (Re < 2000) or turbulent (Re > 4000).
64
A U-tube contains two **immiscible fluids** of densities 𝜌1 and 𝜌2. If fluid 1 is in the left arm and fluid 2 is in the right arm, find the height difference ℎ between the two columns at equilibrium.
## Footnote Equal pressure at the fluid interface implies hydrostatic balance: ρgh is the same on both sides.
65
# True or False: In a steady, incompressible flow, the **pressure gradient** is always **perpendicular** to the flow velocity.
False ## Footnote While the pressure gradient often has components perpendicular to the flow in practical scenarios, it is not constrained to be perpendicular in all cases. The pressure gradient drives changes in **flow velocity** and can have a component parallel to the flow.
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