A box of books weighing 337 N is shoved across the floor by a force of 467 N exerted downward at an angle of 35° below the horizontal.
(a) If μk between the box and the floor is 0.56, how long does it take to move the box 4.00 m starting from rest?
2.518 s
(b) What is the maximum coefficient of kinetic friction between the box and the floor that allows the box to move from this applied force.
To solve this problem, we will use Newton's second law of motion and the equation for frictional force.
(a) To find the time it takes to move the box, we can start by calculating the net force acting on the box in the horizontal direction. The net force can be found by subtracting the force of friction from the applied force.
Step 1: Calculate the force of friction (Ff) using the equation Ff = μk * Fn, where μk is the coefficient of kinetic friction and Fn is the normal force.
Given:
μk = 0.56
Fn = weight of the box = 337 N
Ff = 0.56 * 337 N
Ff = 188.72 N
Step 2: Calculate the net force (Fnet) by subtracting the force of friction from the applied force.
Fnet = Fa - Ff
Fnet = 467 N - 188.72 N
Fnet = 278.28 N
Step 3: Use Newton's second law of motion, F = ma, to find the acceleration (a) of the box.
Since there is no vertical acceleration, we only consider the horizontal component of the applied force.
Fnet = ma
a = Fnet / m
Given:
mass of the box (m) = ? (not given)
Unfortunately, we don't have the mass of the box. Without the mass, we cannot calculate the time it takes to move the box.
(b) To find the maximum coefficient of kinetic friction, we will use the same values as in part (a) and solve for μk.
Ff = μk * Fn
μk = Ff / Fn
Given:
Ff = 188.72 N
Fn = 337 N
μk = 188.72 N / 337 N
μk = 0.559 (rounded to three decimal places)
Therefore, the maximum coefficient of kinetic friction that allows the box to move from this applied force is approximately 0.559.