Motion Under Central Forces

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Questions and Answers

Explain why the cross product of the position vector and force vector is zero for motion under central forces.

Because the force vector is always directed along the position vector in central force motion, making them parallel. The cross product of two parallel vectors is zero.

What is the implication of the total angular momentum being constant in a central force system?

Resultant torque is zero.

In polar coordinates, a particle's position is given by $\vec{r} = r \hat{r}$. Express the velocity vector components ($v_r$, $v_\theta$) in terms of $r$, $\dot{r}$, $\theta$, and $\dot{\theta}$.

$v_r = \dot{r}$ and $v_\theta = r \dot{\theta}$

Relate areal velocity to angular momentum.

<p>Areal velocity is one-half times $r^2$ times angular velocity, which in turn relates to angular momentum.</p> Signup and view all the answers

What is the equation of trajectory?

<p>$d^2u/d\theta^2 + u = -mF(r)/J^2u^2$</p> Signup and view all the answers

State Kepler's first law.

<p>Every planet moves in an elliptic orbit with the sun at one of its foci.</p> Signup and view all the answers

State Kepler's second law.

<p>A line joining a planet and the sun sweeps out equal areas during equal intervals of time.</p> Signup and view all the answers

For a circular orbit, what is the relationship between the period $T$ and the radius $r$?

<p>$T^2 = Kr^3$</p> Signup and view all the answers

What is the general formula for calculating centripetal force of a planet.

<p>$F = m \omega^2 r$?</p> Signup and view all the answers

What is the conclusion drawn by Newton on the centripetal force acting on the planet?

<p>The centripetal force acting on the planet is always directed from planet towards sun.</p> Signup and view all the answers

What did Newton say about the relationship between force and the distance from the planet to the sun?

<p>Force is inversely proportional to the square of the distance from planet to sun.</p> Signup and view all the answers

According to Newton, how is force related to the mass of the planet?

<p>Force is directly proportional to the mass of the planet.</p> Signup and view all the answers

According to Newton, how does force act between two mass particles?

<p>The force not only acts between the planet and the sun but also acts between any two mass particles.</p> Signup and view all the answers

What is the formula for gravitational force between two masses?

<p>$F = (Gm_1m_2)/r^2$</p> Signup and view all the answers

Flashcards

Motion in a Plane

The cross product of position and force vectors.

Central Force

A force directed along the line joining two particles, magnitude depends only on the distance between them.

Linear Momentum (p)

The product of the particle's mass and velocity vector.

Angular Momentum (L)

The cross product of the position vector and the linear momentum vector.

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Conservation of Angular Momentum

Angular momentum is constant when torque is zero.

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Kepler's Second Law

States that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

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Kepler's First Law

Planets move in elliptical orbits with the Sun at one focus.

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Kepler's Third Law

Square of the period is proportional to the cube of the semi-major axis.

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Motion Under Central Forces

Motion where the force acting on an object is always directed towards a central point.

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Constant Areal Velocity

Areal velocity in orbit is constant.

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Newton's Law of Gravitation

Force is proportional to the product of masses and inversely proportional to the square of the distance.

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Study Notes

  • Motion under central forces is being discussed

Key Facts

  • F = kx(vector) = + F(r)r(vector)
  • Fgr = (GmM/r^2)r(vector)
  • FE = (1/4pie0) (q1q2/r^2)r(vector)
  • F(s) = kr(vector)

Motion in a Plane

  • r(vector) x F(vector) = r(vector) x F(r)r(vector) = + F(r) (r(vector) x r(vector)) = 0
  • L = r(vector) x p(vector) = constant
  • F = m * d^2r(vector) / dt^2
  • r(vector) x (d^2r(vector) / dt^2) = 0
  • d/dt (r(vector) x dr(vector)/dt) = dr(vector)/dt x dr(vector)/dt + r(vector) x d^2r(vector)/dt^2 = 0

Parameter h

  • h = r(vector) x dr(vector)/dt = constant
  • h is a parameter and a constant vector
  • r(vector) is a radial vector
  • dr(vector)/dt is a velocity of radial vector

Angular Momentum

  • Angular momentum of a particle moving under a central force is discussed
  • L(vector) = r(vector) * p(vector) where p(vector) = mv(vector)
  • L(vector) = r(vector) x mv(vector) = r(vector) x m(dr(vector)/dt)
  • L(vector) = (r(vector) x dr(vector)/dt) * m = m*h(vector)
  • dL(vector)/dt = mdh(vector)/dt = 0
  • Total angular momentum is constant with time
  • Resultant torque is also zero

Equation of Motion

  • Equation of motion of a particle moving under a central force is discussed
  • r(vector) = x(i) + y(j) = rcos(theta)i + rsin(theta)j
  • r(vector)dot = -sin(theta)i + cos(theta)j, r(vector).rdot = 0

Polar Coordinates

  • r(vector) = cos(theta)i + sin(theta)j
  • r(vectordot) = -sin(theta)i + cos(theta)j
  • r(vector)dot.r(vector)dot = 0
  • From equation, x = rcos(theta) and y = rsin(theta)
  • vx = dx/dt
  • vx = d(rcos(theta))/dt
  • vx = (dr/dt)cos(theta) + rd(cos(theta))/dt
  • vx = (dr/dt)cos(theta) + r(-sin(theta))*(d(theta)/dt)
  • vx = (dr/dt)cos(theta) - rsin(theta)(d(theta)/dt)
  • vx = r(dot)*cos(theta) - rsin(theta)*theta(dot)
  • vy = r(dot)*sin(theta) + rcos(theta)*theta(dot)

Acceleration

  • ax = r(double dot)cos(theta) + r(dot)(-sin(theta))*theta(dot) - r(dot)*sin(theta)*theta(dot) - rcos(theta)*theta(dot)^2
  • ax = r(double dot)cos(theta) - 2r(dot)*theta(dot)*sin(theta) - rcos(theta)*theta(dot)^2 - rsin(theta)*theta(double dot)
  • ay = r(double dot)*sin(theta) + r(dot)*cos(theta)*theta(dot) + r(dot)*cos(theta)*theta(dot) - rsin(theta)*theta(dot)^2 + rcos(theta)*theta(double dot)
  • ay = r(double dot)sin(theta) + 2r(dot)*cos(theta)*theta(dot) - rsin(theta)*theta(dot)^2+ rcos(theta)*theta(double dot)
  • a = ax(i) + ay(j)
  • a = (r(double dot) - rtheta(dot)^2) * (cos(theta)i + sin(theta)j) + (rtheta(double dot) + 2*r(dot)*theta(dot)) *(-sin(theta)i + cos(theta)j)
  • a =(r(double dot) - rtheta(dot)^2)r(vector) + (rtheta(double dot) + 2*r(dot)*theta(dot))*theta(vector)

Newton's Second Law

  • F = ma(vector)
  • For a central force, F(r) = F(r)r(vector)
  • F(r) = m(r(double dot) - r*theta(dot)^2)
  • The first equation of motion: m(rtheta(double dot) + 2r(dot)*theta(dot)) = 0
  • The second equation of motion: mrtheta(double dot) + 2mr(dot)*theta(dot) = 0
  • d/dt (mr^2 theta(dot)) = m2r*r(dot)theta(dot) + mr^2theta(double dot) = r(dot) * (2mr(dot)theta(dot) + mrtheta(double dot)) = 0

Law of Conservation of Angular Momentum

  • m(rtheta(double dot) + 2r(dot)*theta(dot)) = 0
  • d/dt * (mr^2theta(dot)) = 0
  • Here, I = mr^2 and d(theta)/dt = w

Moment of Inertia

  • Now, d/dt * (Iw) = 0
  • dI/dt = 0
  • I = Constant
  • For a small angle, theta = l/r
  • N -> Q
  • PQ perpendicular OQ

Area

  • Area(POQ) = 1/2 * r * r d(theta)
  • dA = 1/2 * r * r d(theta)
  • dA = 1/2 * r^2 d(theta)
  • Rate of change of area is studied
  • dA/dt = 1/2 * r^2 d(theta)/dt = 1/2 * r^2 * theta(dot) which is the areal velocity

Areal Velocity

  • dA/dt = L / 2m = constant
  • dA/dt = I/2m = constant, which is Kepler's second law

Equation of Trajectory

  • In polar coordinates Angular momentum J = mr^2(d(theta)/dt)
  • d(theta)/dt = J/mr^2
  • d(theta)/dt = J/mr^2 * (d(theta)/dt)
  • d/dt = J/mr^2 * d/d(theta)
  • F(r) = m (r(double dot) - r*theta(dot)^2)
  • F(r) = m(J^2/m^2r^2 * d/d(theta) * (1/r^2dr/d(theta)) - r(J/mr^2)^2)
  • F(r) = (J^2/mr^2 d/d(theta) * (1/r^2dr/d(theta)) - J^2/mr^3
  • For r = r(theta)
  • Here, d/d(theta) * (1/r) = -1/r^2 * dr/d(theta)

Equation of trajectory for moving particles

  • d/d(theta) * (1/r) = -1/r^2 * dr/d(theta)
  • F(r) = J^2/mr^2 * [d/d(theta) * (d/d(theta) * (1/r)) - J^2/mr^3]
  • F(r) = -J^2/mr^2 * [d^2/d(theta)^2 * (1/r) + 1/r]
  • Let u = 1/r therefore u = u(theta)
  • d^2u/d(theta)^2 + u = -mF(r) / J^2u^2
  • So, this is equation of trajectory for moving particles under a central force
  • If u = e^-theta, Find trajectory equation
  • du/d(theta) = -e^-theta
  • d^2u/d(theta)^2 = e^-theta = u
  • d^2u/d(theta)^2 + u = -mF(r) / J^2u^2
  • d^2u/d(theta)^2 +(-mF(r) / J^2u^2)
  • 2u = (-mF(r) / J^2u^2)
  • F(r) = (-2J^2u^3)/m = (-2J^2e^-3(theta))/m

Constant Trajectory Equation

  • a =constant
  • r = 2acos(theta), Find trajectory equation.
  • We know that u = 1/r so, u = 1/2acos(theta)
  • u = 1/2a * sec(theta)
  • du/d(theta) = 1/2a * sec(theta)*tan(theta)
  • d^2u/d(theta)^2 = 1/2a * [sec(theta)^3 + tan^2(theta)*sec(theta)]
  • d^2u/d(theta)^2 = 1/2a * sec(theta) * [sec^2(theta) + tan^2(theta)]
  • d^2u/d(theta)^2 = u * [sec^2(theta) + tan^2(theta)]
  • d^2u/d(theta)^2 + u = (-mF(r))/ J^2u^2
  • u[sec^2(theta) + tan^2(theta)] + u = (-mF(r))/ J^2u^2
  • u[sec^2(theta) + tan^2(theta) + 1] = (-mF(r))/ J^2u^2
  • u[2sec^2(theta)] = (-mF(r))/ J^2u^2
  • F(r) = (-u^3/m) * [2sec^2(theta)] * J^2

Kepler's Laws of Planetary Motion

  • First law - Every planet moves in an elliptic orbit and the sun is at focus
  • Second law - A1 = A2 if the time interval is the same. The area swept per second by the line joining the planet to the sun is always constan
  • dA/dt = constant
  • Third law - The square of the time period of the planet is directly proportional to the cube of the semi-major axis of the orbit: T^2 is proportional to a^3

Newtonian conclusion from Kepler's law

  • Third law - T^2 is proportional to a^3
  • For a circular path: T^2 is proportional to r^3
  • T^2 = Kr^3
  • T = sqrt(Kr^3)
  • Where k is a proportionality constant
  • Angular velocity of planet w:
  • w = 2pi/T
  • w = 2pi/sqrt(Kr^3)
  • Due to this Centripetal force of planet is:
  • F= mv^2/r ; v = rw
  • F= m(r^2w^2)/r
  • F= mw^2r

Centripetal Force

  • F= m((2pi/sqrt(Kr^3))^2)r
  • F= (m4pi^2r)/ (Kr^3)
  • F= (4pi^2m)/ (Kr^2)
  • Hence on the basis of Kepler's third law Newton draws the following conclusions:
      1. Centripetal force acting on the planet is always from the planet towards the sun.
      1. This force is inversely proportional to the square of the distance from the planet to the sun i.e. F proportional (1/r^2)
      1. This force is directly proportional to the mass of the planet i.e. F proportional m
      1. Newton generalizes this force and said that this force not only acts between the planet and the sun but also acts b/w any two mass particles.
  • Fproportional m and F proportional 1/r^2
  • F proportional (m1*m2)/r^2
  • F = (Gm1m2)/r^2
  • F(vector) = - (Gm1m2)/r^2 * r(vector)

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