What is the acceleration of a particle at x=10 m if the mass of the particle is 0.2 kg ? the particle is moving along x axis with the potential energy U(x)= 8x^2-2x^4
Work is given by the product of the force and distance. The work done contributes to an increase in potential energy.
Work = -delta U.
Force* distance = -delta U
Here we have the function of the potential energy given as U(x) = 8*x^2 - 2*x^4.
For an infinitesimally small change in distance
d/dx (Force* distance) = -U'(x)
Force = -U'(x)
U'(x) = 16x - 8x^3
At x = 10,
U'(x) = 16*10 - 8*1000
=> 160 - 8000
=> Force = - (-7840)
=> Force = 7840
So the force at this point is 7840.
Force is the product of the mass and acceleration. We know that the mass is 0.2 kg.
Therefore the acceleration is 7840 / 0.2 = 39200 m/s^2
The acceleration is the ratio of the force and the unit mass.
Since we know the potential energy, we'll determine the intensity that is the same as acceleration.
dU/dx = d(8x^2-2x^4)/dx
dU/dx = 16x - 8x^3
Now, we'll substitute x by 10 m:
T = dU/dx
T = 16*10 - 8*10^3
T = 160 - 8000
T = a = -7840 m/s^2