Frequency of small oscillation of ball hanging in accelerating car

December 18, 2015Classical Mechanicshideki

Problem:

A car accelerating at uniform rate $a_o$ has a small ball hung under the roof by a string of length L. What is the angle $\theta$ between the string and the vertical direction when the ball is is steady state? What will be the frequency of oscillation if the ball is set in small oscillation?

Solution:

Steps:

Specify the chosen reference frame
Sketch the free body diagram
Sum the forces along x and along y
Solve for theta

– Frequency of small oscillation of ball hanging in accelerating car

Amount of water that would make the system most stable

December 17, 2015Classical Mechanicshideki

Problem:

An empty cylindrical container has height $L$ , mass $m$ and inner radius $r$ . The center of gravity of the container is located at height h from its bottom. How much water (in terms of height) should one pour into the container to make it most stable? The density of water is $\rho$ .

Solution:

The mass of water adds up to the stability of the system but too much water would make the center of mass high (in terms of height). You need basic knowledge on:

Solving differential calculus – quotient rule
Solving quadratic equation
Finding center of mass

Steps:

Find the center of mass of the system (cylinder + water)
Minimize the center of mass in terms of height of water
Solve for height of water

Try to solve the problem first. If you’re hopeless, then here it is

pset1-3 – height of water with most stable system

Rod hanging by a string next to wall

December 15, 2015Classical Mechanicshideki

Problem:

A uniform rod AB with length $l$ and weight $W$ is placed as shown in the figure, with end A touching a rough wall and end B hanging by a strong, inextensible string. The other end of the string is fixed at point C on the wall is $\mu$ . The rod is hung horizontally and the angle between the string and the rod is $\theta$ . Derive the condition that $\mu$ and $\theta$ should satisfy such that the rod stays at equilibrium.

Solution:

Steps:

Sketch the free body diagram
Sum forces along x
Sum forces along y
Sum torque
Solve system of equations for $\theta$ and $\mu$

Try to solve the problem first without looking at the solution, if you’re still helpless,

Here it is:

pset1-2 – Rod hanging by a string

Acceleration of Missile

December 14, 2015Classical Mechanicshideki

Problem:
Imagine a tank traveling with constant speed $v_1$ along the direction AB, and a missile is chasing the tank with constant speed $v_2$ . The direction of motion of the missile is always pointing towards the tank. At time t the tank is at position F and the missile is at position D, where FD is perpendicular to AB and FD=L. What is the acceleration of the missile at this instant?

There’s a lot of solutions to this problem. But I will provide the formal one (long method). This method requires knowledge skills on:

Differential Calculus
Vector Algebra
Distance-Velocity-Acceleration knowledge

Steps would include:

Writing initial conditions for missile and tank
Write the velocity of tank as magnitude*direction
Differentiate the velocity to get acceleration of missile
Substitute the initial conditions
Simplify
Answer should be $a=\frac{v_1 v_2 }{L} \hat{x}$

Try to solve the problem first without looking at the solution. If you’re still helpless, then here it is:

pset1-1 – acceleration of missile

Position vector for the center of mass from an arbitrary origin

June 19, 2015Classical Mechanicshideki

PROBLEM:
1.2 Goldstein Classical Mechanics 3rd ed.
Prove that the magnitude R of the position vector for the center of mass from an
arbitrary origin is given by the equation

$\displaystyle M^2R^2 = M\sum_i m_i r_i^2 - \frac{1}{2}\sum_{i,j} m_i m_j r_{ij}^2$

SOLUTION:
The position vector for the center of mass is given by Continue reading Position vector for the center of mass from an arbitrary origin →

Derivation of Kinetic energy differential equations

June 18, 2015Classical Mechanicshideki

PROBLEM:
1.1 Goldstein Classical Mechanics 3rd ed.
Show that for a single particle with constant mass the equation of motion
implies the following differential equation for the kinetic energy:

$\displaystyle \frac{dT}{dt} = \bf F \cdot \bf v$

while if the mass varies with time the corresponding equation is

$\displaystyle \frac{d(mT)}{dt} = \bf F \cdot \bf p$

SOLUTION:
The kinetic energy T is given by

$\displaystyle T = \frac{1}{2}m\bf v^2$

The time derivative of kinetic energy is

$\displaystyle \frac{dT}{dt} = \frac{d}{dt} (\frac{1}{2}m\bf v^2)$

chain rule

$\displaystyle \frac{dT}{dt} = m\mathbf{v}\cdot\frac{d\bf v}{dt}$

commutative property of dot product

$\displaystyle \frac{dT}{dt} = m\frac{d\bf v}{dt} \cdot \bf v$

$\displaystyle \frac{dT}{dt} = m\bf a \cdot \bf v$

Newton’s second law

$\displaystyle \frac{dT}{dt} = \bf F \cdot \bf v$

When the mass varies with time, again the kinetic energy T is given by

$\displaystyle T = \frac{1}{2}m\bf v^2$

Multiply both sides by m

$\displaystyle mT = \frac{1}{2}m^2\bf v^2$

$\displaystyle mT = \frac{1}{2}{(m\bf v)}^2$

$\displaystyle mT = \frac{1}{2}\bf p^2$

The time derivative is

$\displaystyle \frac{d(mT)}{dt} = \frac{d}{dt} \frac{\mathbf p^2}{2}$

$\displaystyle \frac{d(mT)}{dt} = \mathbf{p} \cdot \frac{d\bf p}{dt}$

$\displaystyle \frac{d(mT)}{dt} = \mathbf{p} \cdot \bf F$

$\displaystyle \frac{d(mT)}{dt} = \bf F \cdot \bf p$

Comment for corrections/suggestions and further questions. Thanks!

Final temp in Carnot cycle

June 9, 2015Statistical Mechanicshideki

PROBLEM: 2 identical bodies of heat capacities $C_o$ independent of temperature are initially at temperatures $T_H$ and $T_C$ . A Carnot cycle is run b/w them until they come to a common temperature $T_f$ . (a) Find $T_f$ in terms of $T_H$ and $T_C$ . (b) Find total work done by this process on the outside world.
Continue reading Final temp in Carnot cycle →

Compute the internal energy of a non-ideal gas – van der waals

June 5, 2015Statistical Mechanicsnon ideal gashideki

Given: $\displaystyle (p + \alpha\frac{n^2}{V^2})(V-nb) = nRT$
Find: internal energy U

SOLUTION:
The internal energy of a van der waals gas is given by

$\displaystyle U_{real} = U_{ideal} + \int_{V_o}^{V_1} T^2 (\frac{\partial}{\partial T}\frac{p}{T})dV$

To solve the integral, we first find p using algebra

$\displaystyle (p + \alpha\frac{n^2}{V^2})(V-nb) = nRT$

$\displaystyle pV - pnb + \frac{\alpha n^2}{V} - \frac{\alpha n^3b}{V} = nRT$

$\displaystyle p(V-nb) = nRT - \frac{\alpha n^2}{V} + \frac{\alpha n^3b}{V}$

Divide both sides by T(V-nb)

$\displaystyle \frac{p}{T} = \frac{nR}{V-nb} - \frac{\alpha n^2}{TV(V-nb)} + \frac{\alpha n^3b}{TV(V-nb)}$

Differentiate wrt T

$\displaystyle \frac{\partial}{\partial T}\frac{p}{T} = \frac{\alpha n^2}{T^2V(V-nb)} - \frac{\alpha n^3b}{T^2V(V-nb)}$

The integral becomes

$\displaystyle = \int_{V_o}^{V_1} T^2 (\frac{\alpha n^2}{T^2V(V-nb)} - \frac{\alpha n^3b}{T^2V(V-nb)}) dV$

$\displaystyle = \int_{V_o}^{V_1} (\frac{\alpha n^2}{V(V-nb)} - \frac{\alpha n^3b}{V(V-nb)}) dV$

$\displaystyle = (\alpha n^2 - \alpha n^3b) \int_{V_o}^{V_1} \frac{dV}{V(V-nb)}$

$\displaystyle = (\alpha n^2 - \alpha n^3b) \int_{V_o}^{V_1} \frac{dV}{V^2} - \int_{V_o}^{V_1} \frac{dV}{Vnb}$

$\displaystyle = (\alpha n^2 - \alpha n^3b) (\frac{1}{V_o} - \frac{1}{V_1} - \frac{1}{nb} ln\frac{V_1}{V_o})$

Thus,

$\displaystyle U_{real} = \frac{3}{2}nR + (\alpha n^2 - \alpha n^3b) (\frac{1}{V_o} - \frac{1}{V_1} - \frac{1}{nb} ln\frac{V_1}{V_o})$

Comment for corrections and further questions. Thanks!

Calculate B2(T) given potential U

June 1, 2015Statistical Mechanicshideki

Problem: Solve for $B_2(T)$ for $U(r) = \frac{\alpha}{r^n}$ where $n>0$ .

SOLUTION:

Use IBP, then transform the remaining integral into a “gamma function.” Continue reading Calculate B2(T) given potential U →

Transition probability when perturbation is sinusoidal

November 17, 2014Quantum MechanicsGriffiths Quantum Mechanics problem 9.15dhideki

Problem 9.15d Griffiths Quantum Mechanics 1st edition
Given $H'=Vcos(\omega t)$ . Show that transitions only occur to states with energy $E_m=E_N\pm \hbar\omega$ , the transition probability is $P_{N\rightarrow m}=|V_{mN}|^2\,\, \frac{sin^2[(E_N-E_m\pm\hbar\omega)t/2\hbar]}{(E_N-Em\pm\hbar\omega)^2}$

SOLUTION:
We start from the wavefunction $C_m (t)$ then substitute the perturbation $H'$ , integrate then determine the transition probability $|C_m (t)|^2$ . We’ll try to solve this step-by-step and down to algebra level (hopefully).