To solve this problem, let's analyze the forces acting on the trunk and calculate the work done by the applied force and gravity.
### Given data:
- Mass of the trunk: \( m = 52.3 \) kg
- Distance moved along the incline: \( d = 5.95 \) m
- Angle of the incline: \( \theta = 28.0^\circ \)
- Coefficient of kinetic friction: \( \mu_k = 0.19 \)
- Acceleration due to gravity: \( g = 9.81 \) m/s²
### Step 1: Identify the forces acting on the trunk
1. **Gravitational Force**
The weight of the trunk acts downward:
\[ W = mg = (52.3)(9.81) = 512.1 \text{ N} \]
The component of gravitational force parallel to the incline:
\[ F_{\text{gravity, parallel}} = mg \sin \theta = (512.1) \sin 28.0^\circ \]
\[ F_{\text{gravity, parallel}} = 512.1 \times 0.4695 = 240.4 \text{ N} \]
The component of gravitational force perpendicular to the incline:
\[ F_{\text{gravity, perpendicular}} = mg \cos \theta = (512.1) \cos 28.0^\circ \]
\[ F_{\text{gravity, perpendicular}} = 512.1 \times 0.8829 = 452.0 \text{ N} \]
2. **Frictional Force**
The normal force is given by:
\[ N = F_{\text{gravity, perpendicular}} = 452.0 \text{ N} \]
The kinetic friction force is:
\[ F_k = \mu_k N = (0.19)(452.0) \]
\[ F_k = 85.9 \text{ N} \]
3. **Applied Force**
The applied force is horizontal. Let \( F_A \) be its magnitude.
Resolving it into components:
- Along the incline: \( F_A \cos \theta \)
- Perpendicular to the incline: \( F_A \sin \theta \)
Since the trunk moves at constant speed, the net force along the incline is zero:
\[ F_A \cos \theta = F_{\text{gravity, parallel}} + F_k \]
\[ F_A \cos 28.0^\circ = 240.4 + 85.9 \]
\[ F_A \times 0.8829 = 326.3 \]
\[ F_A = \frac{326.3}{0.8829} \]
\[ F_A = 369.7 \text{ N} \]
### Step 2: Work Done by the Applied Force
Work is given by:
\[
W_A = F_A d \cos \theta
\]
\[
W_A = (369.7)(5.95) \cos 28.0^\circ
\]
\[
W_A = (369.7)(5.95)(0.8829)
\]
\[
W_A = 1942.8 \text{ J}
\]
### Step 3: Work Done by Gravity
\[
W_g = - F_{\text{gravity, parallel}} d
\]
\[
W_g = - (240.4)(5.95)
\]
\[
W_g = -1430.4 \text{ J}
\]
### Final Answers:
(a) Work done by the applied force: **1942.8 J**
(b) Work done by the force of gravity: **-1430.4 J**
