The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 45 rpm.

(a) Find the angular speed of the wheel sprocket.
rad/min

(b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket. Round your answer to one decimal place.)
mi/h

(a) Well, if we want to find the angular speed of the wheel sprocket, we need to know how fast it's spinning. And we can figure that out by looking at the pedal sprocket's rotation speed. So let's do the math!

The pedal sprocket has a radius of 4 in., which means its circumference is 2π(4) = 8π in.

Since the cyclist pedals at 45 rpm (revolutions per minute), the pedal sprocket completes 45 revolutions every minute.

So, the pedal sprocket travels a distance of 8π inches in one revolution. And since it does this 45 times in a minute, it travels a total distance of 360π inches each minute.

Since the wheel sprocket is connected to the pedal sprocket with a chain, it travels the same distance and completes the same number of revolutions.

Therefore, the angular speed of the wheel sprocket is also 45 rpm.

(b) Now, to find the speed of the bicycle, we need to take into account the wheel's radius. The wheel has a radius of 13 in.

Knowing the angular speed of the wheel sprocket and the wheel's radius, we can use the equation v = ωr to find the speed of the bicycle, where ω is the angular speed and r is the wheel's radius.

In this case, the angular speed is 45 rpm, which is equivalent to (45 × 2π) rad/min.

So, v = (45 × 2π) rad/min × 13 in.

Calculating this, we get v ≈ 1179.7 in/min.

Now, to convert this to miles per hour, we need to do a bit of unit conversion.

There are 60 minutes in an hour and 5280 feet in a mile, so

v = (1179.7 in/min) × (1 ft/12 in) × (1 mi/5280 ft) × (60 min/1 h).

Doing the math, we get v ≈ 7.27 mi/h.

So, the speed of the bicycle is approximately 7.3 mph.

(a) To find the angular speed of the wheel sprocket, we need to convert the pedal speed (rpm) to radians per minute.

Since there are 2π radians in one revolution, we can calculate the angular speed as follows:

Angular speed of the pedal sprocket = 45 rpm * 2π rad/rev = 90π rad/min

Since the chain connects the pedal sprocket to the wheel sprocket, the angular speeds of both sprockets are equal. Therefore, the angular speed of the wheel sprocket is also 90π rad/min.

(b) To find the speed of the bicycle, we'll use the relationship between linear speed, angular speed, and radius:

Linear speed = Angular speed * Radius

The linear speed of the wheel sprocket is the same as the linear speed of the wheel.

So, the speed of the bicycle is given by the formula:

Speed = Angular speed of wheel sprocket * Radius of wheel

Speed = (90π rad/min) * (13 in.)

Now, we need to convert the units from inches per minute to miles per hour.

1 mile = 5280 feet and 1 foot = 12 inches.

So, 1 mile = 5280 * 12 = 63360 inches.

We also need to convert minutes to hours.

1 hour = 60 minutes.

Therefore, we can calculate the speed of the bicycle as follows:

Speed = (90π rad/min) * (13 in.) * (63360 in./mile) * (1 mile/5280 ft.) * (1 ft./12 in.) * (1 hour/60 min.)

Simplifying the units, we get:

Speed = (90π * 13 * 63360) / (5280 * 12 * 60) miles per hour

Calculating this expression gives us the speed of the bicycle.

To find the angular speed of the wheel sprocket, we need to convert the given pedal speed from revolutions per minute (rpm) to radians per minute.

(a) Angular speed is given by the formula:
Angular speed = 2π × rpm

Substituting the given values:
Angular speed = 2π × 45 rpm

Calculating this:
Angular speed = 90π radians per minute

Therefore, the angular speed of the wheel sprocket is 90π rad/min.

Now, to find the speed of the bicycle, we can use the relationship between the angular speed, radius, and linear speed.

(b) The formula relating angular speed, radius, and linear speed is:
Linear speed = radius × angular speed

Substituting the given values:
Linear speed = 13 in × 90π rad/min

To convert the linear speed to miles per hour, we need to make some unit conversions:
1 mile = 63360 inches
1 hour = 60 min

So the conversion factor is:
63360 inches / 1 mile × 1 hour / 60 min

Calculating the linear speed and conversion:
linear speed = 13 in × 90π rad/min × (63360 / (1 × 60)) (mi/h)

Simplifying this expression:
linear speed = 13 × 90π × 63360 / (60) (mi/h)

Evaluating this expression:
linear speed = 2π × 13 × 6336 (mi/h)

Calculating this expression:
linear speed ≈ 49049.94 (mi/h)

Therefore, the speed of the bicycle is approximately 49049.94 mi/h, rounded to one decimal place.

angular velocity:

1 rotation = 2π radians
45 rotations = 90π radians

angular velocity = 90π radians/minute

Since circumference is linear and we are told the radius of the pedal sprocket is twice that of the wheel sprocket, for every turn of the pedal, the wheel will make 2 rotations
So in 1 minute the wheel turns 90 times
circumference of wheel = 2π(13) = 26π inches
distance covered in 1 minute = 90(26π) = 2340π inches

speed = 2340π inches/min
there are 12(5280) inches in 1 mile
and 60 minutes in 1 hour

speed = 2340π(60/(12(5280)) miles/hr
= appr 7.0 mph