Chapter 1.

Question 1

Give the mathematical definitions of the following: 𝐯 (velocity), 𝐩 (momentum), 𝐅 (force), 𝐚 (acceleration) in terms of 𝐫 (position), t (time), and m (mass).

Answer 1

1. 𝐯 = d𝐫/dt
2. 𝐩 = m𝐯
3. 𝐅 = d𝐩/dt = d/dt (m𝐯) [= m d𝐯/dt when m is independent of time]
4. 𝐚 = d²𝐫/dt²

Question 2

What is an inertial reference frame?

Answer 2

One in which 𝐅 = d𝐩/dt holds true.

Question 3

What is the conservation theorem for the linear momentum of a particle?

Answer 3

If total force |𝐅| = 0, then d𝐩/dt = 0 and linear momentum, 𝐩, is conserved.

Question 4

Define 𝐍 (moment of force / torque) and 𝐋 (angular momentum) mathematically in terms of 𝐫 (position), 𝐩 (momentum), 𝐅 (force).

Answer 4

1. 𝐋 = 𝐫 × 𝐩

2. 𝐍 = 𝐫 × 𝐅

Question 5

Derive the relationship between 𝐍 and 𝐋.

Answer 5

𝐫 × 𝐅 = 𝐍 = 𝐫 × d/dt (m𝐯)
⇔ d/dt (𝐫 × m𝐯) = 𝐫 × m𝐯 + 𝐫 × d/dt (m𝐯)
⇔ 𝐍 = d/dt (𝐫 × m𝐯) = d𝐋/dt

Question 6

What is the conservation theorem for the angular momentum of a particle?

Answer 6

If |𝐍| = 0, then d𝐋/dt = 0 and 𝐋 is conserved.

Question 7

Derive W₁₂ = T₂ - T₁, where T denotes the kinetic energy of the particle at times 1 and 2

Answer 7

W₁₂ = ∫(1 to 2) 𝐅 ⋅ d𝐬 = m ∫(1 to 2) d𝐯/dt ⋅ 𝐯 dt = m/2 ∫(1 to 2) |v|² dt = m/2 (v₁² - v₂²) = T₂ - T₁

Question 8

A force field (and the system) is conservative if _____

Answer 8

W₁₂ is the same for any possible two paths between 1 and 2 such that ∮𝐅 ⋅ d𝐬 = 0

Question 9

Physically, a system cannot be conservative if there is a presence of _____

Answer 9

friction or other dissipative forces

Question 10

What is potential energy in terms of ∮𝐅 ⋅ d𝐬 = 0?

Answer 10

𝐅 = -∇V(𝐫)

Question 11

What is the differential path length for V(𝐫)?

Answer 11

𝐅 ⋅ d𝐬 = -dV ⇔ 𝐅_s = -∂V/∂𝐬

Question 12

What is the energy conservation theorem for a particle?

Answer 12

If the forces acting on a particle are conservative, then T

+ V, or the total energy of the particle, is conserved.

Question 13

For a system where F is dependent on the gradient of a scalar function depending on position and time, Work done for a distance ds is then given by _____, which means V + T is _____ [conserved or not conserved?].

Answer 13

1. 𝐅 ⋅ d𝐬 = -∂V/∂𝐬 ds

2. Not conserved

Question 14

The equation of motion for a system of particles with internal and external forces is _____, and from this equation, the center of mass of the system is derived as follows: _____. We also get linear momentum as _____.

Answer 14

1. For particle i, Σ 𝐅_(ji) + 𝐅ᵉ_i = d𝐩_i/dt
2. d²/dt² Σ_i m_i 𝐫_i = Σ_i 𝐅ᵉ_i + Σ_(i,j, j≠i) 𝐅_(ji) ⇔ 𝐑 - (Σ m_i 𝐫_i)/(Σ m_i) = (Σ m_i 𝐫_i)/M
3. 𝐏 = Σ m_i d𝐫_i/dt = M d𝐑/dt

Question 15

What is the conservation theorem for linear momentum of a system of particles?

Answer 15

If the total external force is zero, the total linear momentum is conserved.