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
Wc = m*g = 2kg * 9.8N/kg = 19.6 N. = Wt.
of crate.
Fc = 19.6N. @ 30 Deg. = Force of crate.
Fp = 19.6*sin30 = 9.8 N.=Force parallel
to incline.
Fv = 19.6*cos30 = 17.0 N. = Force
perpendicular to incline.
Fap = 20 + 40 + 10 = 70 N. = Force applied.
Fn = Fap - Fp - Fk
Fn = 70 - 9.8 - 0.2*17 = 6.4 N. = Net
force.
a = Fn/m = 6.4 / 2 = 3.2 m/s^2.
NOTE: It is assumed the 3 forces act
parallel to and up the plane.
Correction: Fn = 56.8 N.
a = 56.8 / 2 = 28.4 N.
To find the acceleration of the crate, we can follow these steps:
Step 1: Resolve the forces parallel and perpendicular to the incline.
Step 2: Calculate the net force parallel to the incline.
Step 3: Calculate the net force perpendicular to the incline.
Step 4: Calculate the frictional force.
Step 5: Calculate the gravitational force parallel to the incline.
Step 6: Calculate the acceleration.
Let's go through each step in detail:
Step 1: Resolve the forces parallel and perpendicular to the incline.
Breaking down F1, F2, and F3 into their components, we get:
F1_parallel = F1 * sin(θ)
= 20N * sin(30°)
≈ 10N
F2_parallel = F2 * sin(φ)
= 40N * sin(60°)
≈ 34.64N
F3_parallel = F3 * cos(θ)
= 10N * cos(30°)
≈ 8.66N
F3_perpendicular = F3 * sin(θ)
= 10N * sin(30°)
≈ 5N
Step 2: Calculate the net force parallel to the incline.
The net force parallel to the incline can be calculated by summing up the individual forces:
Net force_parallel = F1_parallel + F2_parallel + F3_parallel
= 10N + 34.64N + 8.66N
= 53.3N
Step 3: Calculate the net force perpendicular to the incline.
The net force perpendicular to the incline is the sum of the perpendicular forces:
Net force_perpendicular = F3_perpendicular
= 5N
Step 4: Calculate the frictional force.
The frictional force can be calculated using the equation:
Frictional force = coefficient of friction * net force perpendicular
= μk * Net force_perpendicular
= 0.2 * 5N
= 1N
Step 5: Calculate the gravitational force parallel to the incline.
The gravitational force parallel to the incline can be calculated using the equation:
Gravitational force_parallel = mass * g * sin(θ)
= 2kg * 9.8m/s² * sin(30°)
≈ 9.8N
Step 6: Calculate the acceleration.
Finally, we can calculate the acceleration using Newton's second law:
Net force_parallel = mass * acceleration
acceleration = Net force_parallel / mass
= (Gravitational force_parallel - Frictional force) / mass
= (9.8N - 1N) / 2kg
= 4.4m/s²
Therefore, the acceleration of the crate is approximately 4.4 m/s².
To find the acceleration of the crate, we need to consider both the forces acting on it and the effects of the incline and friction.
First, let's calculate the gravitational force acting on the crate.
Force due to gravity (Fg) = mass (m) * acceleration due to gravity (g)
Fg = 2 kg * 9.8 m/s^2
Fg = 19.6 N
Next, let's resolve the applied forces F1, F2, and F3 into their components along the incline and perpendicular to the incline.
For F1 = 20N:
The component along the incline (F1_parallel) is given by:
F1_parallel = F1 * sin(θ)
F1_parallel = 20N * sin(30°)
F1_parallel = 10 N
The component perpendicular to the incline (F1_perpendicular) is given by:
F1_perpendicular = F1 * cos(θ)
F1_perpendicular = 20N * cos(30°)
F1_perpendicular = 17.32 N
Similarly, for F2 = 40N:
F2_parallel = F2 * sin(φ)
F2_parallel = 40N * sin(60°)
F2_parallel = 34.64 N
F2_perpendicular = F2 * cos(φ)
F2_perpendicular = 40N * cos(60°)
F2_perpendicular = 20 N
And for F3 = 10N:
F3_parallel = F3 * sin(φ) (since φ is the same as in F2)
F3_parallel = 10N * sin(60°)
F3_parallel = 8.66 N
F3_perpendicular = F3 * cos(φ) (since φ is the same as in F2)
F3_perpendicular = 10N * cos(60°)
F3_perpendicular = 5 N
Now, let's calculate the net force acting on the crate along the incline.
Net Force (F_net) = F1_parallel + F2_parallel + F3_parallel - Fk
F_net = 10 N + 34.64 N + 8.66 N - Fk
Fk is the force due to kinetic friction, which can be calculated using the coefficient of kinetic friction (μk) and the perpendicular force applied to the crate.
Fk = μk * F_perpendicular
Fk = 0.2 * (F1_perpendicular + F2_perpendicular + F3_perpendicular)
Fk = 0.2 * (17.32 N + 20 N + 5 N)
Fk = 0.2 * 42.32 N
Fk = 8.464 N
So, the net force along the incline is:
F_net = 10 N + 34.64 N + 8.66 N - 8.464 N
F_net = 44.836 N
Finally, we can calculate the acceleration along the incline using Newton's second law (F = ma), where F is the net force and m is the mass of the crate.
F_net = m * a
44.836 N = 2 kg * a
Solving for a:
a = 44.836 N / 2 kg
a = 22.418 m/s^2
So, the acceleration of the crate along the incline is 22.418 m/s^2.