To calculate the acceleration of the 10.0 kg box being pulled by a 40.0 N force at a 30.0-degree angle above the horizontal, we first need to analyze the forces acting on the box.
1. **Determine the forces acting on the box:**
The applied force can be broken down into its horizontal and vertical components.
- Horizontal Component (Fx):
Fx = F * cos(θ) = 40.0 N * cos(30°) = 40.0 N * (√3/2) ≈ 34.64 N - Vertical Component (Fy):
Fy = F * sin(θ) = 40.0 N * sin(30°) = 40.0 N * 0.5 = 20.0 N
2. **Calculate the normal force (N):**
The weight of the box (W) is equal to mg = 10.0 kg * 9.8 m/s² = 98 N.
Since the vertical component of the applied force reduces the normal force:
N = W – Fy = 98 N – 20.0 N = 78 N
3. **Calculate the frictional force (Ffriction):**
Using the coefficient of kinetic friction (μk) of 0.30:
Ffriction = μk * N = 0.30 * 78 N = 23.4 N
4. **Calculate the net force (Fnet):**
The net force acting on the box can be calculated by subtracting the frictional force from the horizontal component of the applied force:
Fnet = Fx – Ffriction = 34.64 N – 23.4 N = 11.24 N
5. **Calculate the acceleration (a):**
Using Newton’s second law (F = ma), we can solve for acceleration:
a = Fnet / m = 11.24 N / 10.0 kg = 1.124 m/s²
Thus, the acceleration of the box is approximately 1.12 m/s².
6. **Calculating the Coefficient of Static Friction (μs):**
For the coefficient of static friction, this typically needs to be determined experimentally. If the static friction is higher than the kinetic friction, a good estimate could be around 1.5 to 2 times the kinetic friction coefficient, but this can vary depending on the materials involved. Therefore, the coefficient of static friction could be estimated as:
μs ≈ 1.0 (but this is just an estimate and should be verified based on specific conditions).
In summary, the box accelerates at about 1.12 m/s² when pulled at a 30-degree angle, and the coefficient of static friction is approximately estimated to be around 0.30 to 0.60.