A crate with a mass of 50 kg has a coefficient of static friction with the floor of 0.8 and a coefficient of kinetic friction of 0.25. If it is pushed with a force of 425 N, how long will it take to move it across a 65 meter horizontal floor?

19N

199 N

To find out how long it takes to move the crate across the floor, we need to determine whether it will be in a static or kinetic friction state.

First, let's calculate the maximum force of static friction (Fstatic) between the crate and the floor using the formula:

Fstatic = coefficient of static friction * Normal force

The normal force (N) is equal to the weight of the crate, which can be calculated using the formula:

N = mass * gravitational acceleration

Assuming the gravitational acceleration is approximately 9.8 m/s^2, we can substitute the values into the formulas:

N = 50 kg * 9.8 m/s^2 = 490 N

Fstatic = 0.8 * 490 N = 392 N

Since the applied force of 425 N is greater than the maximum force of static friction (392 N), the crate will start moving and enter the kinetic friction state.

Now, let's calculate the force of kinetic friction (Fkinetic) using the formula:

Fkinetic = coefficient of kinetic friction * Normal force

Fkinetic = 0.25 * 490 N = 122.5 N

The force applied to overcome kinetic friction is the net force (Fnet) acting on the crate and is given by:

Fnet = applied force - force of kinetic friction

Fnet = 425 N - 122.5 N = 302.5 N

To find the acceleration of the crate (a), we can use Newton's second law of motion:

Fnet = mass * acceleration

Rearranging the equation, we have:

acceleration = Fnet / mass

acceleration = 302.5 N / 50 kg = 6.05 m/s^2

Now, we need to determine the time it takes for the crate to move 65 meters. We can use the following kinematic equation:

distance = initial velocity * time + 0.5 * acceleration * time^2

The initial velocity (V0) can be assumed to be zero since the crate starts from rest.

Plugging in the known values, we have:

65 m = 0 * t + 0.5 * 6.05 m/s^2 * t^2

Rearranging the equation, we get a quadratic equation:

0.5 * 6.05 m/s^2 * t^2 = 65 m

3.025 m/s^2 * t^2 = 65 m

t^2 = 65 m / 3.025 m/s^2

t^2 = 21.4876 s^2

t = sqrt(21.4876 s^2) ≈ 4.63 s

Therefore, it will take approximately 4.63 seconds to move the crate across the 65 meter horizontal floor.