The front wheel of a bike has a diameter of 92cm and is traveling at a speed of 15mi/hr

Question

The front wheel of a bike has a diameter of 92cm and is traveling at a speed of 15mi/hr

A) Find the angular speed of the wheels

B) How far will the bike travel in 5 minutes

C) How long will it take to travel 60 miles

Answer 1

To solve these problems, let's first convert all units to a consistent system. We'll use the metric system for simplicity.

Given:

1. Diameter of the front wheel (D) = 92 cm
2. Speed of the bike (v) = 15 mi/hr

We'll also need to know some conversion factors: 1 mile = 1.60934 km 1 hour = 3600 seconds

A) Angular Speed (ω) can be calculated using the formula: ω=r/v​ where:

r is the radius of the wheel, which is half of the diameter (46 cm).

r=92/2=46 cmr=292​=46cm

Converting speed from miles per hour (mi/hr) to meters per second (m/s): v=15×1.60934×10003600v=15×1.60934×36001000​ w≈6.7056 m/sv≈6.7056m/s

Now, calculating angular speed: w=6.705646=466.7056​ w≈0.1458 rad/sω≈0.1458rad/s

B) Distance traveled in 5 minutes: First, let's convert 5 minutes to seconds: 5 minutes=5×60 seconds=300 seconds5minutes=5×60seconds=300seconds

The distance traveled (s) can be calculated using the formula: s=rθ where:

1. rθ is the angular displacement, given by θ=ωt, where t is the time.

θ=0.1458×300θ=0.1458×300 θ ≈43.74 radθ≈43.74rad

Now, calculating the distance: s=46×43.74s=46×43.74 s≈2013.24 cms≈2013.24cm

C) Time to travel 60 miles: First, let's convert 60 miles to meters: 60×1.60934×1000≈96,560 m60×1.60934×1000≈96,560m

Now, we can use the formula s=v/t to find the time (t): t=v/s​ =965606.7056t=6.705696560​ ≈14405.89 secondst≈14405.89seconds

Converting this to hours for convenience: =14405.893600t=360014405.89​ ≈4.002 hourst≈4.002hours