A particle of mass 1.1 kg moves under the influence of a potential
U(x) = a/x + bx (a=3 and b=2.5). The particle's motion is restricted to the region x>0.
the force acting on the particle is a/x^2 - b
the point of equilibrium of the particle is 1.10 m
a) What is the minimal work required to move the particle from the equilibrium point to 5.48 m?
b) The particle is released from the rest at 5.48 m. What is the minimal distance from x=0 that it will reach? and what is its maximum velocity?
To find the minimal work required to move the particle from the equilibrium point to 5.48 m, we need to calculate the work done against the force acting on the particle.
a) Work (W) can be calculated using the formula:
W = ∫F(x) dx
Since the force acting on the particle is given by F(x) = a/x^2 - b, we can substitute the values of a and b to get:
F(x) = 3/x^2 - 2.5
To find the minimal work, we can integrate the force from the equilibrium point (x = 1.10 m) to the final point (x = 5.48 m):
W = ∫[1.10 to 5.48] (3/x^2 - 2.5) dx
Integrating this expression will give us the minimal work required.
b) To find the minimal distance from x = 0 that the particle will reach and its maximum velocity, we need to analyze the potential energy and kinetic energy of the particle.
The potential energy (PE) of the particle can be obtained from the given potential function U(x) = a/x + bx:
PE(x) = 3/x + 2.5x
To find the minimal distance from x = 0, we need to find the point where the potential energy is minimum. We can differentiate PE(x) with respect to x and set it equal to zero:
dPE(x) / dx = -3/x^2 + 2.5 = 0
Solving this equation will give us the minimal distance.
The kinetic energy (KE) of the particle can be calculated using:
KE = (1/2)mv^2
To find the maximum velocity, we need to consider conservation of mechanical energy. The mechanical energy (E) of the particle is conserved, so the sum of its potential and kinetic energies will be constant:
E = PE + KE
Using this equation, we can find the maximum velocity by substituting the potential energy function and solving for v.
Keep in mind that these calculations involve integration and solving equations, so actual numerical values need to be substituted to obtain the final results.