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A bicyclist notes that the pedal sprocket has a radius of rp = 9.5 cm while the wheel sprocket has a radius of rw = 6.5 cm. The two sprockets are connected by a chain which rotates without slipping. The bicycle wheel has a radius R = 65 cm. When pedaling the cyclist notes that the pedal rotates at one revolution every t = 1.1 s. When pedaling, the wheel sprocket and the wheel move at the same angular speed.

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MathPhys

Explanation:

Since the chain isn't slipping, the pedal sprocket and wheel sprocket have the same linear velocity.

First, we find the angular velocity of the pedal sprocket:

1.1 rev/s × 2π rad/rev = 2.2π rad/s ≈ 6.9 rad/s

Next, we find the linear velocity of the pedal sprocket:

v = ωr = (6.9 rad/s)(9.5 cm) = 66 cm/s

Now we find the angular velocity of the wheel sprocket:

ω = v/r = (66 m/s) / (6.5 cm) = 10. rad/s

So the linear velocity of the bike is:

v = ωr = (10. rad/s) (0.65 m) = 6.6 m/s

The linear velocity of the bike is proportional to the angular velocity of the pedal.  So if the bicyclist wanted to move at a different speed, say 5.5 m/s, we could find the new angular velocity of the pedal by writing a proportion:

1.1 rev/s / 6.6 m/s = x / 5.5 m/s

x = 0.92 rev/s

Or, 1/x = 1.1 seconds per revolution.

okpalawalter8

Complete Question

A bicyclist notes that the pedal sprocket has a radius of rp = 9.5 cm while the wheel sprocket has a radius of rw = 6.5 cm. The two sprockets are connected by a chain which rotates without slipping. The bicycle wheel has a radius R = 65 cm. When pedaling the cyclist notes that the pedal rotates at one revolution every t = 1.1 s. When pedaling, the wheel sprocket and the wheel move at the same angular speed.

If the cyclist wanted to travel at a speed of v2 = 5.3 m/s, how much time, in seconds, should elapse as the pedal makes one complete revolution?

We have that for the Question, it can be said that  time, in seconds,that should elapse as the pedal makes one complete revolution is t=0.1126

From the question we are told

A bicyclist notes that the pedal sprocket has a radius of rp = 9.5 cm while the wheel sprocket has a radius of rw = 6.5 cm. The two sprockets are connected by a chain which rotates without slipping. The bicycle wheel has a radius R = 65 cm. When pedaling the cyclist notes that the pedal rotates at one revolution every t = 1.1 s. When pedaling, the wheel sp rocket and the wheel move at the same angular speed.

Generally the equation for the time  is mathematically given as

[tex]t=\frac{2\pi rp}{v}\\\\Therefore\\\\t=\frac{2\pi 0.095}{5.3}\\\\t=0.1126[/tex]

Therefore

time, in seconds,that should elapse as the pedal makes one complete revolution is t=0.1126

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