A 23.0 kg box is released on a 39.0° incline and accelerates down the incline at 0.268 m/s2. Find the friction force impeding its motion.
To find the friction force impeding the motion, we can use the equation:
Friction force (Ff) = normal force (N) * coefficient of friction (μ)
To start, let's find the normal force acting on the box. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case:
Normal force (N) = weight (W) * cos(angle)
The weight of the box (W) can be calculated using the formula:
Weight (W) = mass (m) * gravitational acceleration (g)
The gravitational acceleration is approximately 9.8 m/s^2.
So, the weight of the box is:
W = 23.0 kg * 9.8 m/s^2 = 225.4 N
Now, let's find the normal force:
N = 225.4 N * cos(39.0°)
To find the friction force impeding the motion of the box, we need to consider the forces acting on it.
1. The weight of the box (mg) can be resolved into two components: one perpendicular to the incline (mg * cosθ) and one parallel to the incline (mg * sinθ), where θ is the angle of the incline (39.0°).
2. The friction force (F_friction) acts in the opposite direction to the motion.
3. The acceleration of the box (a) is given as 0.268 m/s².
We can use Newton's second law of motion to solve for the friction force:
ΣF = ma
Summing the forces acting in the x-direction (parallel to the incline):
F_friction - mg * sinθ = ma
Now, let's plug in the known values:
F_friction - 23.0 kg * 9.8 m/s² * sin(39.0°) = 23.0 kg * 0.268 m/s²
Since sin(39.0°) = 0.6293, we can simplify the equation:
F_friction - 143.7 N = 6.164 N
Rearranging the equation to solve for the friction force:
F_friction = 6.164 N + 143.7 N
F_friction = 149.864 N
Therefore, the friction force impeding the motion of the box is approximately 149.864 N.