Revision Note 10: Simple Harmonic Motion

Conditions of Simple Harmonic Motion (SHM): Force is (a) restoring and (b) proportional to the displacement from the equilibrium position.

 

This solution applies to all SHM. The different quantities (ω, φ) can be found from initial conditions and A = maximum displacement.

Other formulas common to all SHM:

PE and KE goes back and forth, with PE + KE = constant.

Defining PE at equilibrium = 0, PE at other points:

Different Physical Systems:

Only k varies.

Spring:

k = Spring constant.

Pendulum: (for small θ)

Restoring force:

Notice mass = m cancels out and the period depends on square root of length.

Tip 10.1:

A vertical spring has the same time period as the equivalent horizontal spring – but its equilibrium position shifts due to mg, by an amount such that:

kδ = mg → δ = mg /k

Tip 10.2: Up-down intuition:

For springs:

k  ↑   T ↓

In words, stiffer spring leads to higher frequency and lower time period.

For pendulums:

L  ↑ T ↑

In words, longer pendulums have more stately swings.

Revision Note 9: Gravitation

Newton’s Law of Gravitation:

F_G = Gm₁m₂/r²

Close to earth’s surface:

F_g = GmM/r_E² = weight of mass

So acceleration due to gravity:

g = F_g /m = GM/r_E² = constant, independent of m

For orbit computations: (circular orbit of radius = R, planet mass = m, Star mass = M)

F_G = GmM/R² = mv²/R

GM = v²R = constant for star system

Using v = ωR and T = 2π/ω,

T² is proportional to R³

Kepler’s Laws:

Law 1: Planet moves in an elliptical orbit, with the star in one of the foci.

Law 2: A line drawn from the star to the planet sweeps out equal areas in equal times.

Law 3: If semi-major axis = R (or radius = R for circular approximation), T²/R³ is constant for a star system.

Gravitational Potential Energy:

Close to earth’s surface:

PE = mgh, PE = 0 at surface

For any distance = r:

U = -GMm/r, U = 0 at infinity

Escape velocity:

Gravitational force inside a shell:

Gravitational force inside a shell is 0, since the mass inside is pulled from all sides.

Gravity inside mines is less than that at the surface. F_G when a particle mass, m, is x distance away from the center of the earth (mass = M, radius = r_E) is:

Problem Solving Tips:

Tip 9.1: Weight in an elevator

Weight on a scale in an elevator with acceleration = a (downwards +ve):

W_elev = m(g –a)

When a = g (free fall), W_elev = 0.

When a is –ve, then W_elev is greater than mg.

Tip 9.2: Accelerometer

If the pendulum makes an angle θ with the vertical:

a = g.tan θ

Revision Note 8: Circular Motion

Correspondence between Circular and Linear quantities:

Quantity	Linear	Circular
Displacement	x	θ
Velocity	v = dx/dt	ω = dθ/dt
Acceleration	a = dv/dt	α = dω/dt
Mass/moment	m	I = Σmr2
Force/torque	F	Τ
2nd law	F = ma	Τ = Iα
Impulse	F.dt	Τ.dt
Momentum	p = mv	L = Iω
Work	dW = F.dx	dW = Τ.dθ
Power (P=dW/dt)	P = F.v	P = Τ.ω
Kinetic Energy	KEl = ½ mv2	KEθ = ½ Iω2

Moment of Inertia:

Moment of inertia is the circular equivalent of mass. It deserves special mention as it is smartly crafted to make the 2^nd law and energy relations turn out to be equivalent.

Unlike mass, moment of inertia doesn’t depend just on the body, but also on the axis of rotation. If r is the radius of rotation, then:

r ↑        I↑↑

Conservation of momentum:

Both linear and circular momentums are conserved.

But there is no concept of the sum of linear and angular momentums being conserved – the two have different dimensions and cannot be added.

Kinetic Energy:

Both KE_l  and KE_θ have the same dimension. Suppose a disk (of mass m) rolls downhill through a vertical drop of h. Then the lost potential energy (mgh) is transferred partially to KE_l  and partially to KE_θ. So,

mgh = KE_l + KE_θ

Further solution – knowing the values of KE_l  and KE_θ – needs more equations.

Radial and Tangential acceleration:

Tangential acceleration = αr

Radial acceleration = ω²r

In the case when there is no angular acceleration, tangential acceleration is 0.

Parallel axis theorem:

I_p = I_cm + Md²

Problem Solving Tips:

Tip 8.1: Bridging Linear and Circular quantities

Most problems ask to relate the linear and circular quantities. The bridges are:

Kinematics Bridge:

x = θr

v = ωr

a = αr

Force-Toque Bridge:

F.r = T

Tip 8.2: Finding moment of inertia

General method is to take small pieces of mass, dm and integrate ½ r².dm. From the problem dm = K.dr, where K is some expression. No bridge is needed.

Tip 8.3: Disk rolling downhill

mgh = KE_l + KE_θ = ½ mv² + ½ Iω²

Then use the bridge: v = ωr

Tip 8.4: Massive pulley

If there are two strings, their tensions are different, and then the bridge equation is:

Torque = (T₂ – T₁).r

Sometimes the bridge v = ωr is needed.

Tip 8.5: Bullet embeds in a door

Initial angular momentum = angular momentum of the bullet = L = Iω = mr² . v/r = mvr.

Revision Note 7: Momentum and Collisions

Momentum:

p = mv

What is important is change in momentum,

F∆t = ∆p = m∆v

Impulse:

F∆t is called the impulse, which is equal to change in momentum.

Use calculus when force varies (generally with time):

Center of mass:

For n different masses, the center of mass (x,y) is:

Problem Solving Tips:

Tip 7.1:

For solving momentum problems, first resolve momentum into two orthogonal axes. Then:

p_i,x = p_f,x

p_i,y = p_f,y

The above apply to all types of momentum problems.

The issue is with the number of unknowns and number of equations. There are now three types:

(a) Apply the principle of conservation of momentum only. Some parts of the final momentum are given – maybe the angle or the magnitude – to provide the extra equations to enable solution.

(b) Completely Elastic. No energy lost.  In this case we have the extra equation:

KE_i = KE_f

Or if springs are involved:

KE_i + PE_i = KE_f + PE_f

(c ) Completely inelastic. No, not all energy is lost, but some definitely is lost. The definition of a completely inelastic collision is that the two objects stick together after collision, so their final velocities are the same:

v_1,f = v_2,f

Broken down into two components:

v_x,1,f = v_x,2,f

v_y,1,f = v_y,2,f

Revision Note 6: Work and Energy

Work done

Work done by a single force, F, which applies through a displacement, d, is given by the scalar product:

W = F.d = Fd cosθ

You need calculus if any one of the following is true:

·        The magnitude of the force varies (generally as a function of distance)

·        The θ varies (generally as a function of distance)

In the Calculus version, add up the work done by the force for small distances (dx) – which requires integration from x = x_i = initial position to x = x_f = final position. In other words:

If multiple forces act on the object then the total work done by all forces together is:

·        Algebraic sum of the work done by each force individually.

·        Work done by the net force.

·        Sum of work done over small distances covering the whole distance (integration method)

Work and Energy

If A applies a force on B and does W work, then B does  –W work on A. This comes from Newton’s third law. The object A also transfers W amount of energy to B, and the object B transfers  –W of energy to A.

Work energy theorem:

W_{all forces} = ∆K

Potential energy theorem:

W_grav = -∆U

If the only force is gravitation (or any conservative force) then W_grav = -∆U = ∆K. In this case,

∆K + ∆U = 0

or K + U = constant, which means for initial and final positions:

K_i + U_i = K_f + U_f

The intuition of conservative forces is that work doesn’t get lost and only gets transferred between two bodies – hence talking about potential energy makes sense. PE won’t make sense in case energy leaks out.

Potential Energy

A natural way to measure gravitation potential energy is to set the gravitational PE to 0 when the separation of bodies is infinity. In that case:

Similar considerations apply to energy of an electron around the nucleus and thus that electron while in orbit has negative energy.

Potential energy of a spring of spring constant k compressed by x is given by:

Finding F_grav(x) from U(x):

Power

If P is the power (scalar), then for a constant force, F:

Problem Solving Tips:

Tip 6.1:

For computing W, since cosθ = cos(-θ) we don’t care if the angle is measured from Force to displacement or vice-versa – we just say that it is the angle between the Force and displacement vectors.

However the work done can be negative or zero – see the following diagram:

Scan It

Tip 6.2:

For work, what is important is the displacement, not the distance (though see Work done by friction below). So work done by a constant force through a round trip is zero since displacement is 0.

Tip 6.3:

For work done by friction, the magnitude of the force F_fr = μN continually adjusts to oppose the motion, so θ=180 always.  So for each dx the work done is W_fr = -F_fr.dx. Total work is W_fr = -F_fr.D where D = distance covered. In this case work depends on distance rather than displacement.

Tip 6.4:

When conservative forces are acting, the work done doesn’t depend on path. The potential energy is dependent on location, not the path taken to it.

Tip 6.5:

The spring has the same potential energy for the same amount of compression or same amount of extension. This is because (x)² = (-x)².

Tip 6.6:

Work done by conservative forces = difference in the potential energy between starting and ending locations.

Revision Note 5: Friction

Frictional force (more correctly, the maximum frictional force) is normal force times coefficient of static or dynamic friction (as the case may be).

F_fr = μN

The normal force (N) that is used to compute frictional force is best computed by considering the force by the surface on the body – in other words you need the free-body diagram of the body. This is the best method to compute the force by the body on the surface, as by Newton’s third law the two forces are numerically the same.

Problem Solving Tips:

Tip 5.1:

What is F_fr  for a ramp of incline θ to horizontal when the coefficient of friction is μ? By a previous tip (Tip 2.3), N = mg.cosθ. So,

F_fr,θ = μN = μ mg.cosθ

Tip 5.2: Up-down intuition:

Relating θ with normal force, N, and frictional force, F_fr,θ:

θ ↑        N ↓      F_fr,θ  ↓

The effect of friction decreases as the ramp becomes more and more vertical.