To tackle this physics problem, we need to break it down into manageable parts. We will calculate the work done by the gravitational force, the frictional force, the applied force, and finally determine the change in kinetic energy of the crate. Let’s go through each part step by step.
Work Done by the Gravitational Force
The work done by the gravitational force can be calculated using the formula:
W = F_d \cdot d \cdot \cos(\theta)
Where:
- W = work done by the gravitational force
- F_d = weight component acting along the incline
- d = distance moved along the incline
- θ = angle between the force and the direction of motion
The weight of the crate (F_d) can be calculated as:
F_d = m \cdot g \cdot \sin(\phi)
Where:
- m = mass of the crate = 15 kg
- g = acceleration due to gravity ≈ 9.81 m/s²
- φ = angle of the incline = 30°
Calculating the weight component:
F_d = 15 kg × 9.81 m/s² × \sin(30°) = 15 kg × 9.81 m/s² × 0.5 = 73.575 N
Now, substituting into the work formula:
W = -F_d \cdot d = -73.575 N × 6.0 m = -441.45 J
The negative sign indicates that the gravitational force is acting opposite to the direction of motion.
Work Done by the Frictional Force
The work done by the frictional force can be calculated similarly:
W_f = F_f \cdot d \cdot \cos(180°)
First, we need to find the frictional force (F_f):
F_f = μ_k \cdot N
Where:
- μ_k = coefficient of kinetic friction = 0.47
- N = normal force
The normal force can be calculated as:
N = m \cdot g \cdot \cos(φ) = 15 kg × 9.81 m/s² × \cos(30°) = 15 kg × 9.81 m/s² × 0.866 = 127.64 N
Now, we can find the frictional force:
F_f = 0.47 × 127.64 N = 60.00 N
Now substituting into the work formula:
W_f = -60.00 N × 6.0 m = -360.00 J
The negative sign indicates that friction opposes the motion.
Work Done by the Applied Force
The work done by the applied force can be calculated as follows:
W_a = F_a \cdot d \cdot \cos(0°)
Where:
- F_a = applied force = 150 N
- d = distance = 6.0 m
Substituting the values:
W_a = 150 N × 6.0 m × 1 = 900 J
This indicates that the applied force does positive work on the crate.
Change in Kinetic Energy of the Crate
The work-energy principle states that the total work done on an object is equal to the change in its kinetic energy:
ΔKE = W_a + W_f + W_g
Substituting the values we calculated:
ΔKE = 900 J - 360 J - 441.45 J = 98.55 J
This means the kinetic energy of the crate increases by 98.55 J as it is pulled up the incline.
Summary of Results
- Work done by gravitational force: -441.45 J
- Work done by frictional force: -360.00 J
- Work done by the applied force: 900 J
- Change in kinetic energy: 98.55 J
By analyzing each force acting on the crate, we can understand how they contribute to the overall motion and energy changes of the system. This systematic approach helps clarify the relationships between forces, work, and energy in physics.
