Consider a 0.63-kg sample of metal at room temperature of 20

Complete Question

A 30 cm cm wrench is employed to loosen a bolt, with force applied 0.30m from the bolt. It requires 60 N to loosen the bolt when force is applied perpendicular to the wrench. How much force would be necessary if the force were applied at a 30-degree angle from perpendicular?

Response:

The strength needed is

Clarification:

According to the problem, we know that

    The length of the wrench is

    The distance from the bolt is  

      The force necessary to loosen the bolt is

       The angle of application is

Generally, the torque required for loosening the bolt is defined as

       

       

       

Now for the bolt to loosen at the torque at 90° must equal that at

Thus, the torque at  is represented mathematically as

         

substituting values gives us

           

           

           

       

The electric flux through the cylindrical surface surrounding the infinite charged wire is given by the formula ∅E = E x 2πrl. To analyze this, we consider an infinitely long straight wire with a uniform linear charge density of λ Cm⁻¹. The electric field at a distance r from this charge can be evaluated using a cylindrical Gaussian surface of radius r and length l, oriented along the wire. Only the curved surface of the cylinder contributes to the total flux since the other surfaces are perpendicular to E.

Answer:

≈ 3077.34

Explanation:

To calculate the flux of F (vector field) across surface S, where

F(x,y,z) = y i − x j + k

and S(u,v) = u cos v i + u sin v j + v k, 0 ≤ u ≤ 2, 0 ≤ v ≤ 5π

We will evaluate the following integral:

Substituting for the surface

x = u cos v

y = u sin v

z = v

Then

F(S(u,v)) = u sin v i - u cos v j + k

The normal vector N is computed as

Where:

N = < cos v, sin v, 0 > X <- u sin v, u cos v, 2v

N = < 2v sin v, -2v cos v, u >

F(S(u,v)).N = < u sin v, -u cos v, >. < 2v sin v, -2v cos v, u >

F(S(u,v)).N = 2uv + u

Thus

≈ 3077.34