To calculate the minimum work needed to push a truck up an incline, we need to determine the gravitational force acting on the truck along the incline and then calculate the work done against this force over the distance moved up the incline.
- Calculate the gravitational force acting along the incline:
The weight of the truck (force due to gravity) \( F_g \) can be calculated as: \[ F_g = m \cdot g \] where:
- \( m = 2222 , \text{kg} \) (mass of the truck)
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)
Let's compute the weight: \[ F_g = 2222 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 21756.42 , \text{N} \]
- Calculate the component of the gravitational force acting along the incline:
The force acting along the incline \( F_{\text{incline}} \) can be calculated using: \[ F_{\text{incline}} = F_g \cdot \sin(\theta) \] where \( \theta = 10^\circ \).
Calculating \( \sin(10^\circ) \): \[ \sin(10^\circ) \approx 0.1736 \]
Now, calculate the force along the incline: \[ F_{\text{incline}} = 21756.42 , \text{N} \cdot 0.1736 \approx 3784.17 , \text{N} \]
- Calculate the work done against this force over the distance:
Work \( W \) done against the gravitational force is given by: \[ W = F_{\text{incline}} \cdot d \] where \( d = 53 , \text{m} \) (the distance moved up the incline).
Using the calculated force: \[ W = 3784.17 , \text{N} \cdot 53 , \text{m} \approx 200,000.81 , \text{J} \]
Thus, the minimum work needed to push the truck 53 m up a 10° incline is approximately: \[ \boxed{200,001 , \text{J}} \]