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23 Nov 2019

Particles (mass = m = 1 kg) move under the influence of a centralforce, Fr(r) = - k r where k = 1 N/m. Each particle moves on acircular orbit. The period of revolution as a function of orbitradius r is T(r) = C r^p.

Determine C and p.

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Reid Wolff
Reid WolffLv2
21 Jan 2019

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

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]



A particle of mass m moves under the influence of a central force.The potential energy function is
V(r) = {
K (a2−r2) /2 if r ≤ 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 φa where the particle crosses theboundary r = a.

***HINT GIVEN***

Use the constants of the motion to analyze the orbit.

Write L = mr^2 dφ/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(αt).
Determine c1, c2, and α.

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

Answer B and C: A satellite with a mass of 330 kg moves in acircular orbit 3.00 107 m above the Earth's surface.
(a) What is the gravitational force on the satellite?
= 99.31 N

(b) What is the speed of the satellite?
(km/s)

(c) What is the period of the satellite?
(h)

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