Question(I) In pedaling a bicycle uphill, a cyclist exerts a downward force of 420 N during each stroke. If the diameter of the circle traced by each pedal is 36 cm , calculate how much work is done in each stroke.

Studdy Solution

STEP 1

1. The cyclist exerts a downward force of 420 N during each stroke.
2. The diameter of the circle traced by each pedal is 36 cm.
3. Work is calculated as the product of force and distance in the direction of the force.
4. The pedal moves in a circular path, so the distance is the circumference of the circle.

STEP 2

1. Convert the diameter to radius and then calculate the circumference of the circle.
2. Calculate the work done using the formula for work.

STEP 3

Convert the diameter to radius. The radius $r$ is half of the diameter.
$r = \frac{36 \text{ cm}}{2} = 18 \text{ cm}$
Convert the radius to meters for consistency in units (since force is in Newtons).
$r = 18 \text{ cm} = 0.18 \text{ m}$

STEP 4

Calculate the circumference of the circle, which is the distance the pedal travels in one complete stroke.
The formula for circumference $C$ is:
$C = 2\pi r$
Substitute the radius into the formula:
$C = 2\pi (0.18 \text{ m})$
$C \approx 2 \times 3.1416 \times 0.18$
$C \approx 1.131 \text{ m}$

STEP 5

Calculate the work done using the formula for work, which is the product of force and distance.
$\text{Work} = \text{Force} \times \text{Distance}$
Substitute the given force and the calculated distance:
$\text{Work} = 420 \text{ N} \times 1.131 \text{ m}$
$\text{Work} \approx 474.42 \text{ J}$
The work done in each stroke is approximately:
$\boxed{474.42 \text{ J}}$

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