Introduction
In this problem, we are asked to calculate the final speed of a crate being pulled up an inclined plane with friction. To solve this, we apply Newton’s laws of motion and the work-energy theorem. We’ll also include a descriptive explanation of the free body diagram (FBD).
Given Data
- Mass of crate, m = 30.0 kg
- Force applied, F = 300 N
- Distance moved, s = 15.0 m
- Angle of incline, θ = 30º
- Coefficient of kinetic friction, μk = 0.225
- Initial speed, u = 0 m/s (starts from rest)
- g = 9.8 m/s²
Step 1: Understanding the Free Body Diagram (FBD)
The forces acting on the crate are:
- Gravitational force (mg) acting vertically downward
- Normal force (N) perpendicular to the plane
- Frictional force (f) acting opposite to the direction of motion (down the incline)
- Applied force (F) up the incline
Frictional force: f = μk × N = μk × mg cos(θ)
Component of weight down the incline: mg sin(θ)
Step 2: Net Work Done
Using the Work-Energy Theorem:
Net Work Done = Change in Kinetic Energy
Let’s calculate the net force along the incline:
F_net = Applied force – frictional force – component of weight
F_net = F – (μk × mg × cosθ) – (mg × sinθ)
F_net = 300 – (0.225 × 30 × cos30º × 9.8) – (30 × sin30º × 9.8)
cos30º ≈ 0.866, sin30º = 0.5
F_net = 300 – (0.225 × 30 × 0.866 × 9.8) – (30 × 0.5 × 9.8)
F_net ≈ 300 – (0.225 × 30 × 8.487) – (147)
≈ 300 – (57.3) – (147) = 95.7 N
Step 3: Work-Energy Theorem
Net Work = F_net × s = 95.7 × 15 = 1435.5 J
Change in K.E. = ½ × m × v² – 0
So, 1435.5 = 0.5 × 30 × v²
v² = (1435.5 × 2)/30 = 95.7
v = √95.7 ≈ 9.78 m/s
Answer
The final speed of the crate after moving 15.0 m is approximately 9.78 m/s.
Free Body Diagram Description
- Weight (mg): Acts vertically downwards.
- Normal force (N): Acts perpendicular to the incline, balancing the perpendicular component of the weight.
- Frictional force: Acts parallel to the incline, in the opposite direction of motion.
- Applied force (F): Acts up the incline.
Conclusion
This problem demonstrates the use of forces, components, and energy concepts in mechanics. By applying the work-energy theorem, we find that the crate reaches a speed of about 9.78 m/s after moving 15 meters up the incline, considering both gravity and frictional resistance.