In a pickup game of dorm shuffleboard, students crazed by final exams use a broom to propel a calculus book along the dorm hallway. If the 3.5 kg book is pushed from rest through a distance of 0.77 m by the horizontal 25 N force from the broom and then has a speed of 1.43 m/s, what is the coefficient of kinetic friction between the book and floor?

Question

In a pickup game of dorm shuffleboard, students crazed by final exams use a broom to propel a calculus book along the dorm hallway. If the 3.5 kg book is pushed from rest through a distance of 0.77 m by the horizontal 25 N force from the broom and then has a speed of 1.43 m/s, what is the coefficient of kinetic friction between the book and floor?

μk =

Answer 1

To find the coefficient of kinetic friction (μk) between the book and the floor, we can use the following formula:

μk = (frictional force)/(normal force)

First, let's calculate the frictional force:

The work done by the broom on the book is given by the force applied (25 N) multiplied by the distance covered (0.77 m):

Work = force * distance
= 25 N * 0.77 m
= 19.25 Joules

This work is equal to the net work done on the book, which is the change in kinetic energy (KE). So, we can find the KE of the book as:

KE = 19.25 J

The kinetic energy of an object is given by the formula:

KE = (1/2) * mass * velocity^2

Substituting the values we have:

19.25 J = (1/2) * 3.5 kg * (1.43 m/s)^2
= 3.5 kg * (1.43 m/s)^2 / 2

Now, solving for (1.43 m/s)^2:

(1.43 m/s)^2 = 19.25 J * 2 / 3.5 kg
= 11 J / 3.5 kg
= 3.143 m^2/s^2

So, the velocity squared is approximately 3.143 m^2/s^2.

Next, we need to calculate the frictional force. The frictional force is given by:

frictional force = μk * normal force

Since the book is on a horizontal floor, the normal force and the gravitational force are equal in magnitude:

normal force = weight of the book
= mass * gravity
= 3.5 kg * 9.8 m/s^2
= 34.3 N

Now, we can find the frictional force:

frictional force = μk * normal force

To find μk, we can rewrite it as:

μk = frictional force / normal force

Plugging in the values, we have:

μk = (frictional force) / (normal force)
= (frictional force) / (34.3 N)

To find the value of the frictional force, let's go back to the kinetic energy equation:

KE = (1/2) * mass * velocity^2

From here, we can use the equation to solve for the frictional force:

KE = frictional force * distance

Solving for the frictional force:

frictional force = KE / distance
= 19.25 J / 0.77 m
= 25 N

Finally, substituting this value into the equation for μk, we get:

μk = (frictional force) / (normal force)
= 25 N / 34.3 N
≈ 0.729

Therefore, the coefficient of kinetic friction between the book and the floor is approximately 0.729.