A box of books weighing 324 N is shoved
across the floor by a force of 527 N exerted
downward at an angle of 35� below the hori-
zontal.
The acceleration of gravity is 9.81 m/s2 .
If μk between the box and the floor is 0.59,
how long does it take to move the box 4.35 m,
starting from rest?
To find the time it takes to move the box a distance of 4.35 m, we can use the equation of motion:
s = ut + (1/2)at^2
where s is the distance, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time.
First, let's find the net force acting on the box. The net force is the difference between the downward force exerted on the box and the force of friction:
Net force = downward force - force of friction
The force of friction can be calculated using the formula:
force of friction = coefficient of kinetic friction * normal force
The normal force is the force exerted by the floor on the box and is equal to the weight of the box (given as 324 N) since the box is on a flat surface.
Now, let's calculate the force of friction:
force of friction = coefficient of kinetic friction * normal force
= μk * weight of the box
= 0.59 * 324 N
= 191.16 N
Next, let's calculate the net force:
Net force = downward force - force of friction
= 527 N - 191.16 N
= 335.84 N
The net force is also equal to the mass of the box multiplied by the acceleration:
Net force = mass * acceleration
335.84 N = mass * acceleration
Since we know the weight of the box, we can calculate the mass using the formula:
weight = mass * gravity
324 N = mass * 9.81 m/s^2
mass = 324 N / (9.81 m/s^2)
= 33 kg
Now, we can calculate the acceleration:
acceleration = net force / mass
= 335.84 N / 33 kg
= 10.18 m/s^2
Finally, we can use the equation of motion to find the time it takes to move the box a distance of 4.35 m:
s = (1/2)at^2
4.35 m = (1/2) * 10.18 m/s^2 * t^2
Simplifying the equation:
2 * 4.35 m = 10.18 m/s^2 * t^2
8.7 m = 10.18 m/s^2 * t^2
Rearranging the equation:
t^2 = 8.7 m / (10.18 m/s^2)
t^2 = 0.8547 s^2
Taking the square root of both sides:
t = √(0.8547 s^2)
t ≈ 0.924 s
Therefore, it takes approximately 0.924 seconds to move the box a distance of 4.35 m, starting from rest.