Shreyas’ Notes

Honors Mechanics

PHYS 111

fall, freshman year

Mechanics: study of a system of particles interacting with each other and/or the environment.

Newtonian mechanics applicable when:

  1. space
    • 3-dimensional: 3 coordinates necessary to specify the position of a particle
    • euclidean: euclid’s axioms satisfied
  2. time
    • universal time: time runs the same way for all bodies regardless of their properties

Units §

We describe the magnitude of physical quantities using standard units.

Dimensions §

Combinations of powers of length, mass, time.

Fg=Gm1m2r2F_g = \frac{G m_1 m_2}{r^2}

G=Fgr2m1m2G = \frac{F_g r^2}{m_1 m_2}

GG small, that’s why a large mass necessary to make FgF_g significant

useful for:

not useful for dimensionless constants.

Vectors §

Vectors change[2] under transformation of coordinate systems. Scalars do not.

Examples §

Notation §

A=Aa^\vec A = A \hat a

A=A|\vec A| = A

Translation doesn’t change vectors

Physical laws can be written as relationships between geometric quantities (vectors) in Newtonian Mechanics.

Unit vectors, vector components §

Unit/base vectors:

component form:

A=Axi^+Ayj^+Azk^\vec A = A_x \hat i + A_y \hat j + A_z \hat k

When coord system changed, unit vectors and components change such that the vector itself is unchanged.

Multiplication by a scalar §

Vector Addition §

Head-to-tail, tail-to-other-head

Subtraction: flip subtrahend, add.

Geometrically, but also component-wise.

Properties §

Vector Multiplication §

Dot product §

AB=ABcosθ\vec A \cdot \vec B = AB \cos \theta

independent of coordinate system

projection of A\vec A | B\vec B in the direction of B^\hat B | A^\hat A multiplied by the magnitude of B\vec B | A\vec A

Cartesian coordinates:

A=Axi^+Ayj^+Azk^\vec A = A_x \hat i + A_y \hat j + A_z \hat k

B=Bxi^+Byj^+Bzk^\vec B = B_x \hat i + B_y \hat j + B_z \hat k

\vec A \cdot \vec B = A_x B_x + A_y B_y + A_z B_z$$ [^3] [^3]: $\hat i \times \hat i = 1$, $\hat j \times \hat j = 1$, $\hat k \times \hat k = 1$ #### Cross product $$\vec A \times \vec B = AB \sin \theta \hat c

c^\hat c perpendicular to the plane spanned by A\vec A and B\vec B in the direction of the right hand rule.

independent of coordinate system

Magnitude of cross product: area of the parallelogram spanned by AA and BB

A×B=B×A\vec A \times \vec B = -\vec B \times \vec A

Cartesian coordinates:

A×B=i^j^k^AxAyAzBxByBz\vec A \times \vec B = \left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k}\\ A_{x} & A_{y} & A_{z}\\ B_{x} & B_{y} & B_{z} \end{array}\right|

Vector differentiation §

r(t)=x(t)i^\vec r (t) = x(t) \hat i

dr(t)dt=dx(t)dti^\frac{d \vec r (t)}{dt} = \frac{d x (t)}{dt} \hat i

Geometrically, rate of change of r(t+δ)r(t)r(t + \delta) - r(t).

Notation: dAdt=A˙\frac{d \vec A}{dt} = \dot{\vec A}

Properties §

Same as properties of regular differentiation, except order matters wherever vectors/vector_derivatives are multiplied.

If dAdtA\frac{d \vec A}{dt} \perp \vec A, then A|\vec A| is a constant.

If vector perpendicular to time-der, but time-der != 0, it’s just rotating with the same magnitude.

General kinematic equations §

v˙(t)=a(t)\dot{\vec v} (t) = \vec a (t)

v(t)=v(t0)+t0ta(t)dt\vec v (t) = \vec v (t_0) + \int_{t_0}^{t} \vec a(t') dt

Uniform[3] circular[4] motion §

r=rcosθi^+rsinθj^\vec r = r\cos \theta \hat i + r\sin \theta \hat j

r(t)=rcosωti^+rsinωtj^\vec r (t) = r\cos \omega t \hat i + r\sin \omega t \hat j

v(t)=rωsinωti^+rωcosωtj^v(t) = -r \omega \sin \omega t \hat i + r \omega \cos \omega t \hat j

v=v=vv=rω|v| = v = \sqrt{v \cdot v} = r \omega

rv=0r \cdot v = 0 (rr is perpendicular to vv)

Period ζ=Cv=2πrv=2πω\zeta = \frac{C}{v} = \frac{2 \pi r}{v} = \frac{2\pi}{\omega}

a=rω2cosωti^rω2sinωtj^=ω2rr^\vec a = - r \omega^2 \cos \omega t \hat i - r \omega^2 \sin \omega t \hat j = - \omega^2 r \hat r

a=r2ω4cos2ωt+r2ω4sin2ωt=rω21=rω2|\vec a| = \sqrt{r^2 \omega^4 \cos^2 \omega t + r^2 \omega ^4 \sin^2 \omega t} = r \omega^2 \sqrt{1} = r \omega^2

Circular (and other) motion is sometimes better described using polar coordinates.

  1. rr: distance from origin
  2. θ\theta: angle from the xx axis
Cartesian Polar
x=rcosθx = r \cos \theta r=x2+y2r = \sqrt{x^2 + y^2}
y=rsinθy = r \sin \theta θ=tan1yx\theta = \tan^{-1} \frac{y}{x}
Intersecting horizontal and vertical lines Concentric circles and radial rays from the origin
i^\hat i and j^\hat j are same at every point r^\hat r and θ^\hat \theta change direction from point to point
r^=cosθi^+sinθj^\hat r = \cos \theta \hat i + \sin \theta \hat j
θ^=sinθi^+cosθj^\hat \theta = - \sin \theta \hat i + \cos \theta \hat j

Unit vectors point along the coordinate lines. Move from one coordinate line to the next.

r^˙=θ˙sinθi^+θ˙cosθj^=θ˙θ^\dot{\hat r} = - \dot{\theta} \sin \theta \hat i + \dot{\theta} \cos \theta \hat j = \dot{\theta} \hat \theta

θ^˙=θ˙r^\dot{\hat \theta} = - \dot{\theta} \hat r

change in r^\hat r is along θ^\hat \theta and vice-versa


r=rr^\vec r = r \hat r

v=r˙=r˙r^+rr^˙=r˙r^+rθ˙θ^\vec v = \dot{\vec r} = \dot{r} \hat r + r \dot{\hat r} = \dot{r} \hat r + r \dot{\theta} \hat \theta

\vec a = \dot{\vec v} = (\dot\dot - r \dot \theta^2) \hat r + (2 \dot r \dot \theta + r \dot \dot) \hat \theta

UCM: r=constr = const, θ=ωt\theta = \omega t, θ˙=ω\dot{\theta} = \omega

r=rr^\vec r = r \hat r

v=rωθ^\vec v = r \omega \hat \theta

a=rω2r^\vec a = - r \omega^2 \hat r

in polar coords, on parallel transporting, vector remains the same but components and unit vectors change.

Translate to origin: only radial components

2r˙θ˙θ^2 \dot{r} \dot{\theta} \hat \theta is non-zero only if trajectory is neither radial nor circular

Newton’s Laws §

particle: an object whose size can be neglected when describing its motion

isolated particle: a particle free from the influence of interactions with other particles

  1. gravitational
  2. electromagnetic

Reference frames

Newton’s laws hold only for inertial reference frames.

Newton’s first law §

Inertial reference frames exist

Inertial reference frame: frames of reference where a particle free from external forces moves with a constant velocity

An accelerated frame of reference is a non-inertial reference frame. In non-inertial frames, we introduce fictitious/pseudo forces so that Newton’s laws apply

Frames moving with a constant vv wrt an inertial frame are also inertial

Earth is only approximately an inertial frame

Newton’s second law §

The rate of change of momentum of a particle is equal to the external, real, unbalanced force acting on it.

F=dpdt=ma\vec F = \frac{d \vec p}{dt} = m\vec a

Newton’s third law §

r˙˙=g=gk^\dot{\dot{\vec r}} = \vec g = - g \hat k

z˙˙(t)=g\dot{\dot{z}}(t) = -g

z(t)=z0+vz0t12gt2z(t) = z_0 + v_{z_0} t - \frac {1}{2}gt^2

Phenomenological forces §

Normal force: A contact force exerted by a surface on a body. always \perp to the surface. EM force arising from the pushback from compressed atoms.

Friction arises from contact between surfaces. Arises from attractive forces between subatomic particles.

FfμNF_f \approx \mu N

μ\mu is the coefficient of static friction. usually <1< 1. dimensionless.

0<Ff<N0 < F_f < N

Centripetal acceleration §

acceleration towards the center. caused by tension (resistance to stretching of atoms)

F=Tr^\vec F = -T \hat r (T is the tension in the rope)

a=ω2lr^\vec a = - \omega^2 l \hat r

F=ma    T=mω2lr^\vec F = m\vec a \implies T = m \omega^2 l \hat r

Geostationary orbit §

Uniform circular motion

T=2πω=24hT=\frac{2\pi}{\omega} = 24h

What is rr in terms of RER_E for the orbit to be geostationary?

FgF_g with gravity formula. Fg=msaF_g = m_sa, aa in terms of ω\omega and rr. Equate, simplify. Plug in ω\omega in terms of 24h24h.

Whirling rope §


What is the tension in terms of the radius?

Derive a diffeq in terms of the distance from the origin.

Element δr\delta r of the rope.

total tension = mass of element * acceleration

Pulleys §

Drag forces §

Act to oppose motion of a solid object through a fluid (e.g. air, water). arises from the third law. EM.

fluid resistance.

Depends on:

Fdrag=f(v)v^F_{\textrm{drag}} = -f(v) \hat v

Simple harmonic motion §

Hooke’s law.

F=kxx^\vec F = -kx \hat x (linear restoring force)

if x>0x>0, spring is stretched. F\vec F pulls back towards x=0x = 0

if x<0x < 0, spring is compressed. F\vec F pushes towards x=0x = 0

x˙˙+kmx=0\dot{\dot{x}}+\frac{k}{m} x = 0

x=Bcos(ω0t)+Csin(ω0t)=Acos(ω0t+ϕ)x = B\cos(\omega_0 t) + C\sin(\omega_0 t) = A\cos(\omega_0t+\phi)

ω02=km\omega_0^2 = \frac{k}{m}

projection of uniform circular motion

Gravity §

force on m1m_1 due to m2m_2: $\vec F_{21} = -\frac{Gm_1 m_2}{r^2} \hat r $

shell theorem:

ball theorem:

Momentum §

Per Newton’s 3rd law, f12=f21\vec f_{12} = - \vec f_{21}

m_1 r_1 … = f_12 and m_2 r_2 … = f_21

For particles interacting with each other but isolated from everything else,

total momentum P\vec {\mathbb{P}} vector conserved for isolated systems. dPdt=0\frac{dP}{dt} = 0

Not isolated: dPdt=Fext\frac{dP}{dt} = F_{ext}

Center of mass: R=mirimiR = \frac{\sum m_i r_i}{\sum m_i}

MR=dPdt=FextM \vec R'' = \frac{d\vec P}{dt} = \vec F_{ext}

The center of mass moves as if:

There may be motion around the center of mass

R=dmrdm=dvprdvpR = \frac{\int dm' \vec r'}{\int dm'} = \frac{\int dv' p \vec r'}{\int dv' p}

2 particle system:

Impulse §

0tFextdt\int_0^t \vec F_{\mathrm{ext}} dt

0tFextdt=0tdPdt=P(t)P(0)\int_0^t \vec F_{\mathrm{ext}} dt = \int_0^t \vec dP dt = \vec P(t) - \vec P(0)

Rockets §

MdvdtudMdt=FextM\frac{d\vec v}{dt} - \vec u \frac{dM}{dt} = \vec F_{\mathrm{ext}}

u\vec u is the speed of the expelled mass wrt the rocket (not wrt the inertial frame)

Energy §

Conservative forces §

F(r)\vec F (\vec r) is said to be conservative if rarbF(r)dr\int_{r_a}^{r_b} \vec F(\vec r)\cdot d\vec r is independent of the path for arb. rar_a, rbr_b.

F(r)F(r) is conservative iff cF(r)dr=0\oint_c F(r)\cdot dr = 0

Conservation of mechanical energy §

potential energy: u(r)=orF(r)dru(\vec r) = - \int_o^{\vec r} \vec F (\vec r)\cdot d\vec r

possible to define iff F(r)\vec F(\vec r) is conservative (uu is a function of r\vec r only).

mechanical energy K+U=EK + U = E is constant as the system evolves. EE is conserved.

only differences in EE are meaningful

In general, for a conservative force F(r)\vec F (\vec r) mvoing some particle from 0 to r\vec r with an applied force F(r)- \vec F (\vec r):

0r(F(r)dr=0rF(r)dr=u(r)\int_0^{\vec r} \left(\vec F (\vec r\right) \cdot d\vec r = - \int_0^{\vec r} \vec F (\vec r)\cdot d\vec r = u(\vec r)

F=u(r)\vec F = -\nabla u(\vec r)

abFdr=abudr=u(b)u(a)\int_{\vec a}^{\vec b} \vec F \cdot d\vec r = - \int_{\vec a}^{\vec b} \nabla u \cdot d\vec r = u(\vec b) - u(\vec a)

Angular Momentum §

L=r×p\vec L = \vec r \times \vec p

τ=r×F\tau = \vec r \times \vec F

τ=dLdt\vec \tau = \frac{d\vec L}{dt}

dLdtτi+τe\frac{d\vec L}{dt} - \tau_i + \tau_e

Lz=IωL_z = I \omega

I=ρ2dmI = \int \rho^2 dm

  1. λdb=hp\lambda_{db} = \frac{h}{p} ↩︎

  2. Coord system changes change the components of a vector. The vector itself doesn’t change. ↩︎

  3. θ\theta changes at constant speed. θ=ωt\theta = \omega t ↩︎

  4. rr is constant ↩︎