Explanation:
Given: uniform electric field E= 82.0 kN/C.
a) charge density σ =ε_0 E.
therefore, 
b)Total charge on each face = σA
q=σA

The axial field is the integration of the field from each element of charge around the ring. Because of symmetry, the field is only in the direction of the axis. The field from an element ds in the ring is
<span>dE = (qs*ds)cos(T)/(4*pi*e0)*(x^2 + R^2) </span>
<span>where x is the distance along the axis from the plane of the ring, R is the radius of the ring, qs is the linear charge density, T is the angle of the field from the x-axis. </span>
<span>However, cos(T) = x/sqrt(x^2 + R^2) </span>
<span>so the equation becomes </span>
<span>dE = (qs*ds)*[x/sqrt(x^2 + R^2)]/(4*pi*e0)*(x^2 + R^2) </span>
<span>dE =[qs*ds/(4*pi*e0)]*x/(x^2 + R^2)^1.5 </span>
<span>Integrating around the ring you get </span>
<span>E = (2*pi*R/4*pi*e0)*x/(x^2 + R^2)^1.5 </span>
<span>E = (R/2*e0)*x*(x^2 + R^2)^-1.5 </span>
<span>we differentiate wrt x, the term R/2*e0 is a constant K, and the derivative is </span>
<span>dE/dx = K*{(x^2 + R^2)^-1.5 +x*[(-1.5)*(x^2 + R^2)^-2.5]*2x} </span>
<span>dE/dx = K*{(x^2 + R^2)^-1.5 - 3*x^2*(x^2 + R^2)^-2.5} </span>
<span>to find the maxima set this = 0, giving </span>
<span>(x^2 + R^2)^-1.5 - 3*x^2*(x^2 + R^2)^-2.5 = 0 </span>
<span>mult both side by (x^2 + R^2)^2.5 to get </span>
<span>(x^2 + R^2) - 3*x^2 = 0 </span>
<span>-2*x^2 + R^2 = 0 </span>
<span>-2*x^2 = -R^2 </span>
<span>x = (+/-)R/sqrt(2) </span>
It helps you understand the concept of what you are predicting
Since sphere is performing pure rolling motion
So here the contact point of sphere with respect to inclined plane is always at rest.
This contact point will not move at any instant and and hence sphere will perform pure rolling.
So here we can say that sphere will have some friction force at the contact point but there is no displacement of that point.
So the work done by this static friction will always ZERO.


So there is no work done by friction on the sphere while it is in pure rolling.
Answer:

Explanation:
Work is the product of force and distance.

We know the force to lift the rock was 50 Newtons. The work was 150 Joules.
- 1 Joule is equal to 1 Newton meters.
- We can convert the units to make the problem simpler later. The work is also 150 Newton meters.

Substitute the values into the formula.

We want to solve for distance, so we must isolate the variable. Divide both sides of the equation by 50 Newtons.

The Newtons will cancel out.

The rock was lifted <u>3 meters.</u>