A bicycle with 18-in.-diameter wheels has its gears set so that the chain has a 6-in. radius on the front sprocket and 3-in. radius on the rear sprocket. The cyclist pedals at 195 rpm.
Find the linear speed of the bicycle in in/min (correct to at least two decimal places)
the gear ratio means that the wheel is turning twice as fast as the pedals.
2 * 190rev/min * 18π in/rev = 21488.49 in/min
over 3 rev per second? That's some fast pedaling. But only about 20 mi/hr, since the wheels are so small.
To find the linear speed of the bicycle in inches per minute, we need to calculate the distance traveled by the bicycle in one minute.
First, we need to find the circumference of the front and rear wheels using their diameters:
Circumference of the front wheel = π × diameter = π × 18 inches
Circumference of the rear wheel = π × diameter = π × 18 inches
Next, we need to find the distance traveled by the bicycle in one revolution of the front and rear wheels.
Distance traveled by the front wheel in one revolution = Circumference of the front wheel = π × 18 inches
Distance traveled by the rear wheel in one revolution = Circumference of the rear wheel = π × 18 inches
Now, we need to calculate the ratio of the distances traveled by the front and rear wheels. This ratio represents the gear ratio.
Gear ratio = Distance traveled by the front wheel / Distance traveled by the rear wheel
Gear ratio = (π × 18 inches) / (π × 3 inches)
Gear ratio = 18 / 3 = 6
The gear ratio tells us that for every revolution of the rear wheel, the front wheel will make 6 revolutions.
Finally, we can calculate the linear speed of the bicycle by multiplying the distance traveled per minute by the gear ratio.
Linear speed of the bicycle = Distance traveled by the rear wheel in one minute × Gear ratio
Linear speed of the bicycle = (Distance traveled by the rear wheel in one revolution × revolutions per minute) × Gear ratio
Linear speed of the bicycle = (π × 18 inches × 195) × 6 inches/minute
Evaluating this expression will give us the linear speed of the bicycle in inches per minute, correct to at least two decimal places.
To find the linear speed of the bicycle, we need to start by finding the rotational speed of the wheels in rpm.
The front wheel has a diameter of 18 inches, so its radius is half of that, which is 9 inches. The circumference of the front wheel is given by the formula C = 2πr, where C is the circumference and r is the radius. Substituting the values, we get C = 2π(9) = 18π inches.
Now, let's find the rotational speed of the front wheel in rpm. Since the cyclist pedals at a rate of 195 rpm, the front sprocket also rotates at the same rate. Therefore, the rotational speed of the front wheel is also 195 rpm.
Next, we need to find the rotational speed of the rear wheel. The ratio of the radii of the front and rear sprockets is 6:3, which simplifies to 2:1. Since the front wheel and the rear wheel are connected by the same chain, they must have the same linear velocity.
The linear velocity of the front wheel is given by the formula V = ωr, where V is the linear velocity, ω is the rotational speed in radians per minute, and r is the radius. Substituting the values for the front wheel, we get V_front = 195 rpm * 18π inches = 3510π inches per minute.
Since the rear sprocket has a radius of 3 inches, the linear velocity of the rear wheel is given by V_rear = ω * 3 inches per minute. Setting the linear velocities of the front and rear wheels equal to each other, we have:
3510π inches per minute = ω * 3 inches per minute
Simplifying, we find:
ω = 3510π / 3 radians per minute ≈ 3657.75 radians per minute
Now that we have the rotational speed of the rear wheel, we can find its linear speed. The linear velocity of the rear wheel is given by the formula V = ωr, where V is the linear velocity, ω is the rotational speed, and r is the radius. Substituting the values for the rear wheel, we get:
V_rear = 3657.75 radians per minute * 3 inches ≈ 10973.25 inches per minute ≈ 10973.25 inches per minute (rounded to two decimal places).
Therefore, the linear speed of the bicycle is approximately 10973.25 inches per minute (rounded to two decimal places).