A light (assume massless) rod of length r is fixed at the origin, and a mass M is attached to the other end, as shown.

The rod is constrained to move in the XY-plane. A pendulum of length l
and mass m attached to M can oscillate in the YZ-plane. Use θ for the angle of
the rod in the XY-plane, and Φ for the angle of the
pendulum in the YZ-plane.

(a) Find Lagrange's equations for the system of the rod and pendulum in terms of
θ and Φ.

(b) Find the normal frequencies and normal modes of vibration for small
oscillations.

Solution:

- Concepts:

Lagrangian mechanics, L = T - U - Reasoning:

We are asked to find the Lagrangian and solve the resulting equations of motions for small oscillations. - Details of the calculation:

TM = ½Mr^{2}(dθ/dt)^{2}, T_{m}= ½m((dx/dt)^{2}+ (dy/dt)^{2}+ (dz/dt)^{2}).

x = r cosθ, y = r sin θ + l sinΦ, z = -l cosΦ.

dx/dt = -r sinθ dθ/dt, dy/dt = r cosθ dθ/dt + l cosΦ dΦ/dt,

dz/dt = + l sinsΦ dΦ/dt,

T_{m}= ½m(r^{2}(dθ/dt)^{2}+ l^{2}(dΦ/dt)^{2}+ 2rl cosθ (dθ/dt) cosΦ (dΦ/dt).

U = -mgl cosΦ

L = ½(M + m)r^{2}(dθ/dt)^{2}+ ½ml^{2}(dΦ/dt)^{2}+ mrl cosθ cosΦ (dθ/dt)(dΦ/dt) + mgl cosΦ

∂L/∂(dθ/dt) = (M + m)r^{2}(dθ/dt) + mrl cosθ cosΦ (dΦ/dt)^{ }d/dt ∂L/∂(dθ/dt) = (M + m)r^{2}d^{2}θ/dt^{2}- mrl[sinθ cosΦ (dΦ/dt)(dθ/dt) + cosθ sinΦ (dΦ/dt)^{2}- cosθ cosΦ d^{2}Φ/dt^{2}].

∂L/∂θ = -mrl sinθ cosΦ (dθ/dt)(dΦ/dt).

∂L/∂(dΦ/dt) = ml^{2}(dΦ/dt) + mrl cosθ cosΦ (dθ/dt)^{ }d/dt(∂L/∂(dΦ/dt)) = ml^{2}d^{2}Φ/dt^{2}- mrl[sinθ cosΦ (dθ/dt)^{2}+ cosθ sinΦ (dΦ/dt)(dθ/dt) - cosθ cosΦ d^{2}θ/dt^{2}].

∂L/∂Φ = -mrl cosθ sinΦ (dθ/dt)(dΦ/dt) - mgl sinΦ.

Equations of motion:

(M + m)r^{2}d^{2}θ/dt^{2}- mrl[cosθ sinΦ (dΦ/dt)^{2}- cosθ cosΦ d^{2}Φ/dt^{2}] = 0.

d^{2}Φ/dt^{2}- (r/l)[sinθ cosΦ (dθ/dt)^{2}- cosθ cosΦ d^{2}θ/dt^{2}] + (g/l) sinΦ = 0.

(b) In the small angle approximation (θ << 1, Φ << 1) we have the equations of motion

(M + m)r^{2}d^{2}θ/dt^{2}+ mrl d^{2}Φ/dt^{2}= 0.

d^{2}Φ/dt^{2}+ (r/l)d^{2}θ/dt^{2}+ (g/l)Φ = 0.

Assume Φ = Φ_{0}exp(iωt), θ = θ_{0}exp(iωt) for a normal mode. Then

ω^{2}(M + m)r^{2}θ_{0}+ ω^{2}mrlΦ_{0}= 0, ω^{2}(Φ_{0}+ (r/l)θ_{0}) - (g/l)Φ_{0}= 0.

There is only one oscillatory solution with ω not equal to zero.

θ_{0}= -(m/(m + M))(l/r)Φ_{0}, ω^{2}= ((m + M)/M)(g/l).

A simple pendulum of length l (massless string) and mass m is suspended from a pivot point on the circumference of a thin massless disc of radius a that rotates with a constant angular velocity ω about its central axis as shown in the figure. Find the equation of motion of the mass m in terms of the generalized variable θ.

Solution:

- Concepts:

Lagrangian Mechanics - Reasoning:

All forces except the forces of constraint are derivable from a potential. The Lagrangian formalism is well suited for such a system. - Details of the calculation:

The coordinates of the pivot point are X = a cosωt, Y = a sinωt.

The coordinates of the mass m are x = X + l sinθ, y = Y - l cosθ.

dx/dt = -aω sinωt + l cosθ dθ/dt, dy/dt = aω cosωt + l sinθ dθ/dt.

T = ½m((dx/dt)^{2}+ (dy/dt)^{2})

= ½m[a^{2}ω^{2}+ l^{2}(dθ/dt)^{2}+ 2aωl(dθ/dt)(-sinωt cosθ + cosωt sinθ)]

= ½m[a^{2}ω^{2}+ l^{2}(dθ/dt)^{2}+ 2aωl(dθ/dt) sin(θ - ωt)].

U = mgy = mg(a sinωt - l cosθ).

L = T - U.

Equation of motion: d/dt ∂L/∂(dθ/dt) - ∂L/∂θ = 0.

∂L/∂(dθ/dt) = ml^{2}(dθ/dt)+ maωl sin(θ - ωt).

d/dt(∂L/∂(dθ/dt)) = ml^{2}d^{2}θ/dt^{2}+ maωl cos(θ - ωt)(dθ/dt - ω).

∂L/∂θ = maωl (dθ/dt)cos(θ - ωt) - mgl sinθ.

ml^{2}d^{2}θ/dt^{2}+ maωl cos(θ - ωt)(dθ/dt - ω) - maωl (dθ/dt)cos(θ - ωt) + mgl sinθ = 0.

d^{2}θ/dt^{2}- (a/l)ω^{2}cos(θ - ωt) + (g/l)sinθ = 0

is the equation of motion.

Consider the pendulum illustrated in the figure with l the
length of the string, m the mass of the ball, F_{g} the gravitational
force, and Φ the angular displacement. (You may assume the string
to be of fixed length and negligible mass). Use Lagrange's
equation to derive an equation of motion that neglects terms of order Φ^{3} and higher.

Solution:

- Concepts:

Lagrange's Equations - Reasoning:

All forces except the forces of constraint are derivable from a potential. The Lagrangian formalism is well suited for such a system. We are asked to find the equation of motion. - Details of the calculation:

T = ½ m l^{2}(dΦ/dt)^{2}, U = -mglcos(Φ), L = T - U.

We have one generalized coordinate, Φ, and one generalized velocity, dΦ/dt.

d/dt(∂L/∂(dq/dt)) - ∂L/∂q = 0.

The equation of motion therefore is ml^{2}dΦ^{2}/dt^{2}+ mglsin(Φ) = 0,

or dΦ^{2}/dt^{2}+ (g/l)sin(Φ) = 0.

sinΦ = Φ - Φ^{3}/3! + ...

Neglecting terms of order Φ^{3}and higher we have dΦ^{2}/dt^{2}+ (g/l)Φ = 0.

A simple pendulum of length with a bob of mass m is attached to a massless
support moving horizontally with constant acceleration a.

(a)
Compute the Lagrangian function.

(b) Write down Lagrange's equation of
motion.

(c) Simplify and compare the equation of motion to
that for a simple pendulum with a fixed (motionless) support.

Solution:

- Concepts:

Lagrange's equations - Reasoning:

We are asked to obtain Lagrange's equation of motion. - Details of the calculation:

(a) L = T - U.

T = ½m((dx/t)^{2}+(dy/dt)^{2}).

x = ½at^{2}+ sinθ, y = -b cosθ.

dx/dt = at + cosθ(dθ/dt), dy/dt = sinθ(dθ/dt).

T = ½m(b^{2}(dθ/dt)^{2}+ a^{2}t^{2}+ 2at cosθ(dθ/dt)).

U = mgy = -mg cosθ.

L = ½m(b^{2}(dθ/dt)^{2}+ a^{2}t^{2}+ 2at cosθ (dθ/dt)) + mg cosθ.

(b) ∂L/∂ (dθ/dt) = mb^{2}(dθ/dt) + mat bcosθ, d/dt(∂L/∂ (dθ/dt)) = mb^{2}d^{2}θ/dt^{2}- mat bsinθ (dθ/dt) + ma bcosθ,

∂L/∂θ = -matbsinθ (dθ/dt) - mg bsinθ.

bd^{2}θ/dt^{2}+ g sinθ + a cosθ = 0.

(c) sin(θ + θ_{0}) = sinθ cosθ_{0}+ cosθ sinθ_{0}.

Let g sinθ = c sinθ cosθ_{0}, and a cosθ = c cosθ sinθ_{0}.

Then g = c cosθ_{0}, and a = c sinθ_{0}, c = (g^{2}+ a^{2})^{½}, tanθ_{0}= a/g.

Define φ = θ + θ_{0}. Then the equation of motion becomes

bdφ^{2}/dt^{2}= -c sinφ. This equation becomes bdθ^{2}/dt^{2}= -g sinθ when a = 0.

For a simple pendulum we have bdθ^{2}/dt^{2}= -g sinθ.

A point particle of mass m slides without friction along a hoop of radius R and mass M. The hoop rolls without slipping along a horizontal surface. What is the frequency of small oscillations of the point mass, when it is close to the bottom of the hoop?

Solution:

- Concepts:

Lagrangian mechanics - Reasoning:

All forces except the forces of constraint are derivable from a potential. The Lagrangian formalism is well suited for such a system. - Details of the calculation:

Let θ be the angle through which the hoop rotates and Φ be the angle the line from the center of the hoop to the particle makes with the vertical as shown below.

For the hoop X = Rθ. For the point mass x = Rθ + R sinΦ, y = -R cosΦ.T = ½(MR

^{2}(dθ/dt)^{2}+ I(dθ/dt)^{2}) + ½m[R(dθ/dt) + R (dΦ/dt) cosΦ]^{2}+ ½m[R (dΦ/dt) sinΦ]^{2 }= (M + ½m)R^{2}(dθ/dt)^{2}+ ½mR^{2}(dΦ/dt)^{2}+ mR^{2}cosΦ(dθ/dt)(dΦ/dt).

U = - mgR cosΦ.

L = T - U.

∂L/∂(dθ/dt) = 2(M + ½m)R^{2}(dθ/dt)+ mR^{2}cosΦ(dΦ/dt) = constant, because (dθ/dt) is cyclic.

∂L/∂θ = 0.

In the small angle approximation we then have

(2M + m)d^{2}θ/dt^{2}+ md^{2}Φ/dt^{2}= 0.

d^{2}θ/dt^{2}= -[m/(2M + m)]d^{2}Φ/dt^{2}.

In the small angle approximation we keep only terms to second order in the small quantities in the Lagrangian or terms to first order in the small quantities in the equations of motion.

L = (M + ½m)R^{2}(dθ/dt)^{2}+ ½mR^{2}(dΦ/dt)^{2}+ mR^{2}(dθ/dt)(dΦ/dt) - mgRΦ^{2}/2.

∂L/∂(dΦ/dt) = mR^{2}(dΦ/dt) + mR^{2}(dθ/dt).

d/dt(∂L/∂(dΦ/dt)) = mR^{2}d^{2}Φ/dt^{2}+ mR^{2}d^{2}θ/dt^{2}.

∂L/∂Φ = -mgRΦ.

mR^{2}d^{2}Φ/dt^{2}+ mR^{2}d^{2}θ/dt^{2}+ mgRΦ = 0.

d^{2}Φ/dt^{2}+ d^{2}θ/dt^{2}+ (g/R)Φ = 0.d

^{2}Φ/dt^{2}-[m/(2M + m)]d^{2}Φ/dt^{2}+ (g/R)Φ = 0.

[2M/(2M + m)]d^{2}Φ/dt^{2}= -(g/R)Φ.

d^{2}Φ/dt^{2}= -ω^{2}Φ

ω^{2}= (g/R)(2M + m)/(2M).

A thin rod of linear density ρ and length L is balanced on a cylindrical arc of radius R as shown.

If one end of the rod is tapped, find the frequency of small oscillations for the rod.

Solution:

- Concepts:

The Lagrangian formalism, small oscillations - Reasoning:

We are asked to find the frequency of small oscillations. - Details of the calculation:

Assume we have a very thin rod.

L = T - U. - T = ½M((dX/dt)
^{2}+ (dY/dt)^{2}) + ½I(dθ/dt)^{2}.

U = MgY.

X and Y are the coordinates of the CM.

Let the origin of the coordinate system be the center of the circle in the figure above.

X = R sinθ - Rθ cosθ.

Y = R cosθ + Rθ sinθ.

Small angle approximation: In L, keep only terms up to second order in θ.

X = Rθ - Rθ = 0.

Y = R(1 - θ^{2}/2) + Rθ^{2 }= R(1 + θ^{2}/2).

dX/dt = 0, dY/dt = Rθ dθ/dt.

Therefore

L = ½MR^{2}θ^{2}(dθ/dt)^{2}+ ½M(L^{2}/12)(dθ/dt)^{2}- MgR(1 + θ^{2}/2)

≈ ½M(L^{2}/12)(dθ/dt)^{2}- Mg R(1 + θ^{2}/2).

Again: In the small angle approximation, where we only keep terms to second order in the small quantities.

The equation of motion is

d^{2}θ/dt^{2}= -(12gR/L^{2})θ,

which yields ω = (12gR/L^{2})^{½}, independent of the mass of the rod.

Consider an Atwood machine
with a massless pulley and two masses, m and M, which are attached at opposite
ends to a string of fixed length that is hung over the pulley. For this
Atwood machine the center of the pulley is supported by a spring of spring
constant k.

(a) Find the Lagrangian and the resulting equations of
motion.

(b) Find the equilibrium position of the pulley and its frequency of
oscillation. Consider your result in the limit m = M and discuss.

Solution:

- Concepts:

Small oscillation, Lagrangian mechanics. - Reasoning

We are asked to find the Lagrangian and the equations of motion. - Details of the calculation:

(a) Assume that the center of the pulley is a distance z measured in the downward direction from the equilibrium length of the spring. Assume that the distance from the center of the pulley for M is x measured in the downward direction. For m the distance from the center of the pulley downward is L - x, where L is a constant related to the length of the string.

The kinetic energy of the two masses is

T = ½M(dz/dt + dx/dt)^{2}+ ½m(dz/dt - dx/dt)^{2}.

The potential energy of the two masses and the spring is

U = -Mg(z + x) - mg(z - x) + ½kZ^{2}.

The Lagrangian is

L = ½M(dz/dt + dx/dt)^{2}+ ½m(dz/dt - dx/dt)^{2}+ Mg(z + x) + mg(z - x) - ½kz^{2 = }½M((dz/dt^{)2}+ (dx/dt)^{2}+ 2(dz/dt)(dx/dt)) + ½m((dz/dt^{)2}+ (dx/dt)^{2}- 2(dz/dt)(dx/dt))

+ Mg(z + x) + mg(z - x) - ½kz^{2}.

The x-equation of motion is

(M - m)g = (M + m)d^{2}x/dt^{2}+ (M - m)d^{2}z/dt^{2}.

The z-equation of motion is

(M + m)g - kz = (M + m)d^{2}z/dt^{2}+ (M - m)d^{2}x/dt^{2}.

(b) Solving for d^{2}x/dt^{2}from the x-equation:

d^{2}x/dt^{2}= (g - d^{2}z/dt^{2})(M - m)/(M + m).

Substituting into the z-equation:

(M + m)g - kz = (M + m)d^{2}z/dt^{2}+ (M - m)^{2}(g - d^{2}z/dt^{2})/(M + m).

[(M + m)^{2}- (M - m)^{2}]d^{2}z/dt^{2}= (M + m)^{2}g - (M + m)kz - (M - m)^{2}g.

4Mm d^{2}z/dt^{2}= 4Mmg - (M + m)kz.

d^{2}z/dt^{2}= g - ((M + m)/(4Mm))kz.

The equilibrium position z_{0}of the center of the pulley requires d^{2}z/dt^{2}= 0. Therefore

z_{0}= (g/k) 4Mm/(M + m).

Let z' = z - z_{0}. Then d^{2}z'/dt^{2}= - ((M + m)/(4Mm))kz'.

z' = Cexp(iωt)

The frequency of oscillation is

ω^{2}= k(M + m)/(4Mm) = k/(4μ). where μ = Mm/(M + m).

In the limit m = M the frequency of oscillation reduces to ω^{2}= k/(2m), which is what you expect since neither mass is accelerating due to gravity.