A 1060 N crate is being pulled across a level floor by a force F of 300 N at an angle of 35° above the horizontal. The coefficient of kinetic friction between the crate and the floor is 0.25. Find the magnitude of the acceleration of the crate in m/s^2
To find the magnitude of the acceleration of the crate, we need to calculate the net force and then use Newton's second law of motion, which states that F = ma (force equals mass times acceleration).
Step 1: Calculate the horizontal and vertical components of the force applied (F).
The vertical component of the force (Fy) can be found using trigonometry:
Fy = F * sin(angle)
Fy = 300 N * sin(35°)
Fy ≈ 170.65 N
The horizontal component of the force (Fx) can also be found using trigonometry:
Fx = F * cos(angle)
Fx = 300 N * cos(35°)
Fx ≈ 245.03 N
Step 2: Calculate the force of kinetic friction (fk).
The force of kinetic friction (fk) can be found by multiplying the coefficient of kinetic friction (μk) by the normal force (N), where N is the force perpendicular to the surface.
N = mg, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s²).
N = 1060 N * 9.8 m/s²
N ≈ 10388 N
fk = μk * N
fk = 0.25 * 10388 N
fk ≈ 2597 N
Step 3: Calculate the net force in the horizontal direction (Fnet).
Fnet = Fx - fk
Fnet = 245.03 N - 2597 N
Fnet ≈ -2351.97 N
Since the friction force opposes the applied force, it has a negative sign.
Step 4: Calculate the acceleration (a).
Using Newton's second law of motion, F = ma, where F is the net force and m is the mass of the crate.
F = ma
-2351.97 N = 1060 kg * a
Solving for a:
a = -2351.97 N / 1060 kg
a ≈ -2.22 m/s²
The magnitude of the acceleration of the crate is approximately 2.22 m/s².