Factor label method with rotational motion

[wallace13]

Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 8.4 rad/s. The wheel has a radius of 0.40 m. If you ride the bike for 2090 s, how far would you have gone if the bike could move?
w= rad/ sec
2 pi= rev
v= m/s
2pi x r= circumference

.4m x 2pi= 2.5 m/ rev
2.5 m/rev x 8n4 rev/ sec = 21 m/s
21m/s x 2090 s = 43890 m

 

[LowlyPion]

Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 8.4 rad/s. The wheel has a radius of 0.40 m. If you ride the bike for 2090 s, how far would you have gone if the bike could move?

w= rad/ sec
2 pi= rev
v= m/s
2pi x r= circumference

.4m x 2pi= 2.5 m/ rev
2.5 m/rev x 8n4 rev/ sec = 21 m/s
21m/s x 2090 s = 43890 m

8.4 is given in radians / sec. There are 2π radians in a revolution.

 

[thomate1]

Using the factor label method, we can calculate the distance traveled by the bike in this scenario. We know that the wheel is rotating at a rate of 8.4 rad/s, which can be represented as w= 8.4 rad/s. We also know that 2pi represents one revolution, so we can convert the rotational speed to revolutions per second by dividing by 2pi, giving us 8.4 rad/s ÷ 2pi = 1.34 rev/s.

Next, we can use the formula v= rw, where v represents linear velocity, r represents the radius of the wheel, and w represents the angular velocity. In this case, the linear velocity is equal to the circumference of the wheel, which can be calculated by multiplying the radius (0.40 m) by 2pi. This gives us a value of 2.5 m/rev. Multiplying this by the rotational speed of 1.34 rev/s gives us a linear velocity of 2.5 m/rev x 1.34 rev/s = 3.35 m/s.

Finally, we can use the formula d= vt, where d represents distance, v represents linear velocity, and t represents time. In this scenario, we are given a time of 2090 seconds, so we can plug in our calculated linear velocity of 3.35 m/s and our time of 2090 seconds to find the distance traveled. This gives us a distance of 3.35 m/s x 2090 s = 43890 m.

Therefore, if the bike could move, you would have traveled 43890 meters in 2090 seconds. This calculation shows the usefulness of the factor label method in solving problems involving rotational motion, as it allows us to convert between different units and accurately calculate the desired quantity.