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8) A particle of mass m moves under the influence of an attractivecentral force f(r).
a) Using the concept of effective potential show that by a properchoice of initial conditions a circular orbit can result. [8]
b) The circular orbit is now subject to a small radialperturbation. Determine the relation that must hold among, r, f(r)and df/dr for this orbit to be stable. [8]
c) Now assume that the force law has the general form: f(r) =-K/r^n. Determine the maximum value of n for which the circularorbit can be stable. [4]

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Lelia LubowitzLv2

19 Sep 2019

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(A) A satellite of mass m is in circular orbit about theearth.
The radius of the orbit is r0 and the mass of the earth isME.
Find the total mechanical energy of the satellite.

(B) Now suppose that the satellite moves in the extreme upper
atmosphere of the earth where it is retarded by a constant
weak frictional force f. The satellite will slowly spiral intoward
the earth. Since the friction force is weak, the change
in radius will be very slow. We can therefore assume that
at any instant the satellite is effectively in a circular orbitof
average radius r. Find the approximate change in radius r
per revolution of the satellite.

(C) Calculate r for these parameters: r = 1.045 RE; f =
\ensuremath{1.300\times 10Ë{-4}} N; m = 91 kg.

(D) Find the approximate change of kinetic energy of the
satellite K per revolution, for the same parameters as (C).
Express the result as the fraction of the current kineticenergy.
Is K positive or negative? Explain briefly.

A particle of mass m moves under the influence of a central force.The potential energy function is
V(r) = {
K (a2âr2) /2 if r â¤ a
0 if r > a

as illustrated in the figure. The initial position is (x,y) = (a/2,0) and the initial velocity is (vx,vy) = (0, v0).
[Data: m = 1.0 kg; K =17.1 N/m; a = 1.0 m; r0 = 0.5 a; v0 = 2.0m/s.]

(A) Determine the time ta when the mass crosses the boundary r =a.

(B) Determine angular position Ïa where the particle crosses theboundary r = a.

***HINT GIVEN***

Use the constants of the motion to analyze the orbit.

Write L = mr^2 dÏ/dt = the initial value.
What is the initial value?

Write E = (1/2)m (dr/dt)^2 + L^2/(2mr2) +V(r )
= the initial value. What is the
initial value?

Part A. In the energy equation,
change the variable from r to u,
where r =sqrt(u). The result can be
solved by u(t) = c1+c2 cosh(Î±t).
Determine c1, c2, and Î±.

Part B.
Ï = Integral 0 to t [L/(mr^2)] dt

Good Vibes: Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there.

(d) move to the left until it reaches x=−A and then begin to move to the right.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged

(b) doubled

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M

(d) L and N

(e) M and P

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

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