To move a large crate across a rough floor, you push on it with a force F at an angle of 21°, below the horizontal, as shown in the figure. Find the acceleration of the crate, given that the mass of the crate is m = 38 kg, the applied force is 329 N and the coefficient of kinetic friction between the crate and the floor is 0.52.
Use the same approach used here. Only the numbers have changed.
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12.6
To find the acceleration of the crate, we need to consider the forces acting on it and use Newton's second law of motion.
1. Resolve the applied force F into its horizontal and vertical components:
Fx = F * cosθ
Fy = F * sinθ
Here, θ = 21°
Fx = 329 N * cos(21°)
= 307.92 N
Fy = 329 N * sin(21°)
= 115.53 N
2. Determine the force of friction acting on the crate:
The force of friction (f) can be calculated using the equation:
f = μ * N
Here, μ = 0.52 (coefficient of kinetic friction)
The normal force (N) can be calculated using the equation:
N = m * g
Here, m = 38 kg (mass of the crate)
g = 9.8 m/s² (acceleration due to gravity)
N = 38 kg * 9.8 m/s²
= 372.4 N
f = 0.52 * 372.4 N
= 193.65 N
3. Determine the net force acting on the crate in the horizontal direction:
The net force (F_net) can be calculated as:
F_net = Fx - f
F_net = 307.92 N - 193.65 N
= 114.27 N
4. Calculate the acceleration of the crate using Newton's second law:
F_net = m * a
Here, a = acceleration of the crate (to be calculated)
114.27 N = 38 kg * a
a = 114.27 N / 38 kg
a = 3 m/s²
Therefore, the acceleration of the crate is 3 m/s².
To find the acceleration of the crate, we need to consider the forces acting on it and apply Newton's second law of motion.
1. Identify the forces acting on the crate:
- The applied force F
- The weight of the crate (mg), where g is the acceleration due to gravity (approximately 9.8 m/s^2)
- The force of kinetic friction (fk)
2. Calculate the force of kinetic friction:
The force of kinetic friction can be calculated using the equation:
fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
The normal force N can be calculated using the equation:
N = mg
where m is the mass of the crate and g is the acceleration due to gravity.
So, N = (38 kg)(9.8 m/s^2) = 372.4 N
Now, we can calculate the force of kinetic friction:
fk = (0.52)(372.4 N) = 193.65 N
3. Resolve the applied force into horizontal and vertical components:
The horizontal component of the applied force is F * cos(θ), where θ is the angle of the force below the horizontal.
The vertical component of the applied force is F * sin(θ).
So, the horizontal component is F * cos(21°) = 329 N * cos(21°) = 305.33 N
The vertical component is F * sin(21°) = 329 N * sin(21°) = 118.62 N
4. Calculate the net force acting on the crate in the horizontal direction:
Since the force of kinetic friction acts in the opposite direction to the applied force, the net force in the horizontal direction is:
Fnet = F(horizontal) - fk
= 305.33 N - 193.65 N
= 111.68 N
5. Apply Newton's second law of motion:
The net force is equal to the mass of the crate multiplied by its acceleration:
Fnet = m * a
Rearranging the equation, we get:
a = Fnet / m
= 111.68 N / 38 kg
= 2.94 m/s^2
Therefore, the acceleration of the crate is 2.94 m/s^2.