Work Energy Theorem: Delta K Calculation for Particle Moving in x Direction

[vak3]

A particle moving in the x direction is being acted upon by a net force F(x)=Cx^2, for some constant C. The particle moves from x initial =L to x final=3L. What is Delta K, the change in kinetic energy of the particle during that time?

I tried thih by doing the integral of F(x), replacing x with 2L (because final-initial, 3L-L). I got the answer 2/3 CL^3, and its telling me I am off by a multiplicative factor. Is this becuase I did the integral wrong, or am I missing something?

Thanks!

 

[Doc Al]

Not sure what you did or what you mean by "replacing x with 2L". Step 1: Find the anti-derivative. Step 2: Evaluate it over the interval x = L to x = 3L. (Evaluate for x = 3L and for x = L and subtract.)

 

[ogb p]

The Work Energy Theorem states that the work done on an object equals the change in its kinetic energy. In this case, the net force acting on the particle is F(x)=Cx^2, and it moves from x initial =L to x final=3L. To calculate the change in kinetic energy, we can use the work-energy equation:

W = ΔK

where W is the work done and ΔK is the change in kinetic energy.

To find the work done, we can use the formula for work:

W = ∫F(x)dx

Since the force is given as F(x)=Cx^2, we can substitute this into the work equation and integrate from L to 3L:

W = ∫Cx^2dx = C∫x^2dx = C(x^3/3)|L to 3L = C[(3L)^3/3 - (L)^3/3] = 2/3CL^3

Therefore, the change in kinetic energy is:

ΔK = 2/3CL^3

It seems that you have done the integration correctly, but you may have forgotten to include the constant C in your final answer. The change in kinetic energy is directly proportional to the net force acting on the particle, so the constant C should be included in the final answer.

In conclusion, the change in kinetic energy of the particle moving in the x direction, acted upon by a net force F(x)=Cx^2 from x initial =L to x final=3L, is 2/3CL^3. Remember to always double check your calculations and include all necessary constants in your final answer.