You are in a room in a basement with a smooth concrete floor (friction force equals 40 N) and a nice rug (friction force equals 55 N) that is 3 m by 4 m. However, you have to push a very heavy box from one corner of the rug to the opposite corner of the rug. Will you do more work against friction going around the floor or across the rug, and how much extra?

I will do more work against friction going around the floor.

ΔW = 5 J

Explanation:

Given

Ff₁ = 40 N

Ff₂ = 55 N

a = 3 m

b = 4 m

We have to get the distance d₁ as follows

d₁ = a + b = 3 m + 4 m = 7 m

And d₂:

d₂ = √ (a² + b²) = √ ((3 m) ² + (4 m) ²)

⇒ d₂ = 5 m

then we use the equations

W₁ = Ff₁*d₁ = (40 N) (7 m) = 280 J (work against friction going around the floor)

W₂ = Ff₂*d₂ = (55 N) (5 m) = 275 J (work against friction going across the rug)

since W₂ < W₁ I will do more work against friction going around the floor.

Now, we apply the formula

ΔW = W₁ - W₂ = 280 J - 275 J = 5 J