newtons second law. find coefficient for kinetic friction

Question

an eraser is slid across the floor. it covers 2.09m in 1.44s. find the coefficient for kinetic friction.

i dont understand how μk is supposed to be found because the mass is not given.

Answer 1

It turns out that you don't actually need the mass of the eraser to determine the coefficient of friction here. You're given that the eraser moved 2.09m over 1.44s while a frictional force was applied on it, so the object is undergoing constant deceleration. You can determine this deceleration by using one of the kinematic equations.

Δx = v_ot + 1/2at²)

Δx = 2.09

t = 1.44

The variable v_o is 0 since the eraser's final velocity was 0. Although v_o denotes initial velocity of the object (which wasn't actually 0), we can assume this value is 0 because whether the object is accelerating from 0 to whatever that velocity was, or decelerating from that velocity to 0 (which is the current scenario), Δx is the same in both cases. Therefore, to make life a bit easier, we assume it's the scenario where the object is accelerating rather than decelerating, that way we can assume that v_o is 0. Therefore, we get.

2.09 = 1/2a(1.44)²

We then solve for a and get a = 2.02. Newton's second law says f = ma, so f = m(2.02). In this case, f represents the frictional force, which we know can also be found from the formula ...

f_fric = µf_n

f_n is the normal force of the object

Therefore we have an equality ...

f_fric = m(2.02) = µf_n.

Normal force is calculated by multiplying the mass of an object by 9.8, which is the acceleration of gravity. Therefore...

m(2.02) = µ(9.8m).

Since m appears on both side of the equation, it cancels out, so we have...

2.02 = µ*9.8

Solving for µ, we get ...

µ = 2.02/9.8 = 0.206

Answer 2

We do not need the mass because it will cancel out, but we need to know the initial speed v₀ of the eraser.

The sliding distance under friction is

d = v₀t + at²/2 = v₀t + (F/m)t²/2 = v₀t - [μ_k(mg)/m]t²/2

d = v₀t - μ_kgt²/2

You know d and t, but we need v₀ to complete the answer. Is there anything else given in the book?