A force has the dependence Fx(x) = -kx^4 on the displacement x, where the constant k = 20.3 N/m^4. How much work does it take to change the displacement, working against the force, from 0.730 m to 1.35 m?
If you mean the spring force is
F(x) = - k x^4
the force you have to exert is the negative of that
then integral from x1 to x2 of
integral-F(x) dx
which is (1/5)k [ x2^5 - x1^5 ]
= (1/5) k [ 1.35^5 - .73^5 ]
Well, let me do some quick calculations while juggling a few numbers here... Ah, got it!
To find the work done by the force, you'll need to integrate the force with respect to displacement. In this case, the work done is given by the formula:
W = ∫[-kx^4]dx
Integrating this function with respect to x gives us:
W = -k∫[x^4]dx
Integrating x^4 gives us (1/5)x^5, so we have:
W = -k(1/5)x^5 + C
Now, plugging in the values for x, we get:
W = -k(1/5)(1.35^5 - 0.730^5)
Just to make it more fun, let's calculate that:
W = -20.3(1/5)(1.35^5 - 0.730^5) Nm
Now, let's see what the answer is. Oh, drumroll, please!
*Drumroll*
W ≈ -8.37 Nm
And there you have it! The work required to change the displacement from 0.730 m to 1.35 m while working against the force is approximately -8.37 Nm. Keep it up, and you'll be a physics expert in no time!
To find the work done against the force, we can integrate the force with respect to displacement over the given interval.
The work done is given by the equation:
W = ∫ F(x) dx
Given that the force has the dependence F(x) = -kx^4 = -20.3x^4, we can integrate this expression over the interval from x = 0.730 m to x = 1.35 m:
W = ∫[-20.3x^4]dx (from 0.730 to 1.35)
To find the definite integral, we can use the power rule of integration:
∫x^n dx = (x^(n+1))/(n+1)
Using this rule, we can integrate:
W = [-20.3 * (1/5) * x^5] (from 0.730 to 1.35)
Now, we can substitute the upper and lower limits:
W = [-20.3 * (1/5) * (1.35^5)] - [-20.3 * (1/5) * (0.730^5)]
Calculating this expression:
W ≈ 3.8102 - 0.4745
W ≈ 3.3357 J
Therefore, it takes approximately 3.3357 Joules of work to change the displacement from 0.730 m to 1.35 m, working against the force.
To find the work done in this scenario, you need to calculate the integral of the force with respect to the displacement. In this case, you have a force that depends on the displacement according to Fx(x) = -kx^4, where k = 20.3 N/m^4.
To calculate the work, you integrate the force with respect to the displacement over the given range. In this case, you are changing the displacement from 0.730 m to 1.35 m.
The formula for calculating work is given by:
Work = ∫ F(x) dx
In this case, the force is F(x) = -kx^4, so the work becomes:
Work = ∫ (-kx^4) dx
To solve this integral, you can use the power rule for integration. The power rule states that the integral of x^n with respect to x is (1/(n+1))x^(n+1). Applying the power rule to our equation, we have:
∫ x^n dx = (1/(n+1)) x^(n+1)
Using the power rule, the integral becomes:
Work = (-k/(4+1)) ∫ x^4 dx
Simplifying further:
Work = (-k/5) ∫ x^4 dx
To solve this definite integral, you can substitute the limits of integration, which are 0.730 m and 1.35 m:
Work = (-k/5) [x^5/5] from 0.730 to 1.35
Calculating the lower and upper limits:
Work = (-k/5) [(1.35^5/5) - (0.730^5/5)]
Finally, substitute the value of k = 20.3 N/m^4:
Work = (-20.3/5) [(1.35^5/5) - (0.730^5/5)] N-m or J
By plugging in the values and evaluating the expression, you can calculate the work done.