you are moving a 50 kg crate by sliding it. you push on the crate with a force of 315 n at 21 degrees below the horizontal. the coefficient of kinetic friction is 0.21. how far does the crate move in 15 seconds?
force down on ground = 50*9.81+315sin 21
so
friction force = .21(50*9.81+315sin 21)
so in the horizontal direction
F=ma= 50 a = 315cos21 -.21(50*9.81+315sin 21)
solve for a
I assume that the initial speed is zero
then
x = (1/2)a t^2 = 112.5 * a
M*g = 50 * 9.8 = 490 N. = Wt. of the crate.
Fn = 490*Cos 0 + 315*sin21 = 602.9 N. = Normal force.
Fk = u*Fn = 0.21 * 602.9 = 126.6 N. = Force of kinetic friction.
Fap-Fk = M*a.
315*Cos21-126.6 = 50a, a = ?.
d = 0.5a*t^2, t = 15s.
To calculate the distance the crate moves, we need to analyze the forces acting on it and apply Newton's laws of motion. Here's how you can find the solution step by step:
1. Resolve the applied force into horizontal and vertical components:
The horizontal component Fx = F * cos(θ) = 315 N * cos(21°)
The vertical component Fy = F * sin(θ) = 315 N * sin(21°)
2. Calculate the force of friction:
The force of kinetic friction Ffriction = coefficient of kinetic friction * Normal force
The Normal force N = mass * gravity = 50 kg * 9.8 m/s^2
Ffriction = 0.21 * (50 kg * 9.8 m/s^2)
3. Determine the net force acting horizontally:
The horizontal force is the difference between the applied force and the force of friction.
Fnet = Fx - Ffriction
4. Apply Newton's second law of motion, F = ma, to find the acceleration:
Fnet = m * a
Solve for acceleration, a = Fnet / m
5. Calculate the distance traveled using the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since the crate starts from rest, the initial velocity is 0.
Now, let's calculate step by step:
1. Resolve the applied force:
Fx = 315 N * cos(21°)
Fy = 315 N * sin(21°)
Fx ≈ 296.8 N
Fy ≈ 111.5 N
2. Calculate the force of friction:
N = 50 kg * 9.8 m/s^2 ≈ 490 N
Ffriction = 0.21 * 490 N ≈ 102.9 N
3. Determine the net force:
Fnet = Fx - Ffriction
Fnet = 296.8 N - 102.9 N ≈ 193.9 N
4. Calculate the acceleration:
a = Fnet / m
a = 193.9 N / 50 kg ≈ 3.88 m/s^2
5. Calculate the distance traveled:
distance = 0 * 15 s + (1/2) * 3.88 m/s^2 * (15 s)^2
distance ≈ 8.73 m
Therefore, the crate will move approximately 8.73 meters in 15 seconds when pushed with a force of 315 N at 21 degrees below the horizontal, considering the coefficient of kinetic friction as 0.21.