To calculate the minimum work required to push a car up an incline, we need to consider the component of the weight of the car acting down the incline. The formula for the work done against gravity when moving an object on an incline is:
\[ W = F_d \cdot d \]
where:
- \( W \) is the work done,
- \( F_d \) is the force acting down the incline,
- \( d \) is the distance moved along the incline.
First, we need to find the weight of the car (\( W_c \)):
\[ W_c = m \cdot g \]
where:
- \( m = 1,755 , \text{kg} \) (mass of the car),
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity).
Calculating the weight of the car:
\[ W_c = 1,755 , \text{kg} \cdot 9.81 , \text{m/s}^2 \approx 17,224.55 , \text{N} \]
Next, we need to find the component of this weight acting down the incline. The force due to gravity acting down the incline (\( F_d \)) can be calculated using the sine of the angle of the incline:
\[ F_d = W_c \cdot \sin(\theta) \]
For a 20° incline:
\[ F_d = 17,224.55 , \text{N} \cdot \sin(20^\circ) \]
Calculating \( \sin(20^\circ) \):
\[ \sin(20^\circ) \approx 0.342 \]
Now substituting this value in:
\[ F_d = 17,224.55 , \text{N} \cdot 0.342 \approx 5,885.86 , \text{N} \]
Now, we can calculate the work done to push the car a distance of 65 m up the incline:
\[ W = F_d \cdot d = 5,885.86 , \text{N} \cdot 65 , \text{m} \approx 382,576.74 , \text{J} \]
Thus, the minimum work needed to push the car up the incline is approximately:
\[ \boxed{382,577 , \text{J}} \]