A particle P of mass m=1 moves on the x-axis under the force field
F=36/(x^3) - 9/(x^2) (x>0).
Show that each motion of P consists of either (i) a periodic oscillation between two extreme points, or (ii) an unbounded motion with one extreme point, depending on the value of the total energy.
Regards
To analyze the motion of the particle P under the given force field, we need to examine the potential energy, the kinetic energy, and the total energy of the system.
First, let's find the expression for the potential energy. The potential energy U(x) is defined as the negative of the integral of the force function over displacement:
U(x) = -∫ F(x) dx
U(x) = -∫ (36/(x^3) - 9/(x^2)) dx
U(x) = -(-12/x^2 + 9/x) + C
U(x) = 12/x^2 - 9/x + C
Next, let's find the expression for the kinetic energy. The kinetic energy K is defined as (1/2)m(v^2), where v is the velocity of the particle. Here, the mass m is given as 1, so the kinetic energy simplifies to:
K(x) = (1/2)v^2
Since the velocity v = dx/dt, we can write:
K(x) = (1/2)(dx/dt)^2
Now, let's find the equation for the total energy E of the system. The total energy E is the sum of the kinetic and potential energies:
E(x) = K(x) + U(x)
E(x) = (1/2)(dx/dt)^2 + 12/x^2 - 9/x + C
We can simplify this expression as follows:
E(x) = (1/2)(dx/dt)^2 + 12/x^2 - 9/x + C
Multiplying by 2:
2E(x) = (dx/dt)^2 + 24/x^2 - 18/x + 2C
Since the energy E is a constant, the expression for 2E(x) must remain constant as well. Therefore, let's denote this constant as a new constant A:
(dx/dt)^2 + 24/x^2 - 18/x + 2C = A
Now, let's consider the two cases depending on the value of the total energy:
(i) Periodic oscillation between two extreme points:
In this case, the total energy E is negative. Let's assume E = -D, where D is a positive constant.
Substituting this into the equation, we get:
(dx/dt)^2 + 24/x^2 - 18/x + 2C = -D
Since D is positive, the term (dx/dt)^2 must be greater than the other terms in magnitude. This means the velocity of the particle must be non-zero. Hence, the particle will oscillate between two extreme points for which dx/dt = 0. These extreme points can be found by solving the equation for x values when (dx/dt)^2 = 0.
(ii) Unbounded motion with one extreme point:
In this case, the total energy E is positive. Let's assume E = D, where D is a positive constant. Substituting this into the equation, we get:
(dx/dt)^2 + 24/x^2 - 18/x + 2C = D
Since D is positive, the term (dx/dt)^2 must be less than the other terms in magnitude. This means the velocity of the particle must be zero at some x = x0 value. As the particle moves away from x0, the velocity and x will increase without bound, resulting in an unbounded motion.
In summary, each motion of particle P under the given force field consists of either a periodic oscillation between two extreme points (for negative total energy) or an unbounded motion with one extreme point (for positive total energy).