A particle is attached to point O from A by the force F=-k/x3. If x=x0 and v=0 when t=0, determine velocity and acceleration as a function of x, and the time it takes to travel from A to O.
To find the velocity and acceleration as a function of x, we can use Newton's second law, which states that the force acting on an object is equal to its mass times its acceleration.
Given that the force acting on the particle is F = -k/x^3, we can equate this to the mass of the particle times its acceleration:
F = -k/x^3 = m * a
Rearranging the equation, we get:
a = (-k/m) * (1/x^3)
Now let's integrate the acceleration to find the velocity as a function of x:
v = ∫ a dx
v = ∫ [(-k/m) * (1/x^3)] dx
Integrating this equation, we get:
v = (-k/m) * [(-2)/(2x^2)]
Simplifying further:
v = (k/m) * (1/x^2)
So, the velocity as a function of x is v = (k/m) * (1/x^2).
To find the time it takes to travel from A to O, we can use the fact that v = dx/dt, where dx is the change in position and dt is the change in time. Rearranging this equation, we get:
dt = dx / v
Integrating both sides of the equation:
∫ dt = ∫ (1/v) dx
The left side yields the time it takes to travel from A to O, which we'll call T.
T = ∫ (1/v) dx
Plugging in the expression for v, we have:
T = ∫ (1/[(k/m) * (1/x^2)]) dx
T = ∫ x^2 / (k/m) dx
Integrating this equation will give us the value of T, the time it takes to travel from A to O, as a function of x.