Three applied forces F1= 20N, F2= 40N and F3= 10N act on a crate with a mass of 2kg on an incline with θ = 30o, φ = 60o and μk = 0.2, find the acceleration

It matters where the forces act, and in what direction.

To find the acceleration of the crate, we will first calculate the net force acting on it by considering the gravitational force, applied forces, and the force of friction.

Step 1: Calculate the gravitational force:
The gravitational force (Fg) can be calculated using the formula:
Fg = m * g
where m is the mass of the crate and g is the acceleration due to gravity. In this case, the mass of the crate is given as 2 kg and the acceleration due to gravity is approximately 9.8 m/s^2.

Fg = 2 kg * 9.8 m/s^2
Fg = 19.6 N

Step 2: Calculate the force parallel to the incline due to F1:
The force F1 can be decomposed into two components: one parallel to the incline (F1p) and one perpendicular to the incline (F1n). The force parallel to the incline can be calculated using the formula:
F1p = F1 * sin(θ)
where θ is the angle of the incline (30o in this case).

F1p = 20 N * sin(30o)
F1p = 10 N

Similarly, calculate the force parallel to the incline due to F2:
F2p = F2 * sin(φ)
where φ is the angle of the incline (60o in this case).

F2p = 40 N * sin(60o)
F2p = 34.64 N

Step 3: Calculate the force of friction:
The force of friction (Ff) can be calculated using the formula:
Ff = μk * Fn
where μk is the coefficient of kinetic friction and Fn is the normal force acting on the crate.

The normal force can be calculated using the formula:
Fn = Fg * cos(θ)
where θ is the angle of the incline (30o in this case).

Fn = 19.6 N * cos(30o)
Fn = 16.96 N

Now, plug in the values for μk and Fn to calculate Ff:

Ff = 0.2 * 16.96 N
Ff = 3.392 N

Step 4: Calculate the net force:
The net force (Fnet) can be calculated by summing the forces acting on the crate:
Fnet = F1p + F2p - Ff

Fnet = 10 N + 34.64 N - 3.392 N
Fnet = 41.248 N

Step 5: Calculate the acceleration:
Finally, we can use Newton's second law of motion to calculate the acceleration (a) of the crate:

Fnet = m * a

41.248 N = 2 kg * a

a = 41.248 N / 2 kg
a ≈ 20.624 m/s^2

Therefore, the acceleration of the crate is approximately 20.624 m/s^2.