Math / Geometry

Question(II) What is the minimum work needed to push a $950-\mathrm{kg}$ car 510 m up along a $9.0^{\circ}$ incline? Ignore friction.

Studdy Solution

STEP 1

1. The car has a mass of $950 \, \text{kg}$ .
2. The incline is $9.0^\circ$ .
3. The distance along the incline is $510 \, \text{m}$ .
4. Friction is ignored.
5. Work is calculated as the force parallel to the incline times the distance.

STEP 2

1. Determine the gravitational force acting on the car.
2. Calculate the component of the gravitational force parallel to the incline.
3. Calculate the work done to move the car up the incline.

STEP 3

Determine the gravitational force acting on the car:
$F_{\text{gravity}} = m \cdot g$
where $m = 950 \, \text{kg}$ and $g = 9.8 \, \text{m/s}^2$ .
$F_{\text{gravity}} = 950 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 9310 \, \text{N}$

STEP 4

Calculate the component of the gravitational force parallel to the incline:
$F_{\text{parallel}} = F_{\text{gravity}} \cdot \sin(\theta)$
where $\theta = 9.0^\circ$ .
$F_{\text{parallel}} = 9310 \, \text{N} \cdot \sin(9.0^\circ)$
Calculate $\sin(9.0^\circ)$ :
$\sin(9.0^\circ) \approx 0.1564$
$F_{\text{parallel}} = 9310 \, \text{N} \times 0.1564 \approx 1456 \, \text{N}$

STEP 5

Calculate the work done to move the car up the incline:
$W = F_{\text{parallel}} \cdot d$
where $d = 510 \, \text{m}$ .
$W = 1456 \, \text{N} \times 510 \, \text{m}$
$W \approx 742560 \, \text{J}$
The minimum work needed to push the car up the incline is:
$\boxed{742560 \, \text{J}}$

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