A bicyclist is riding at 27 km/hour. She is pedaling at 95 rpm. Her rear wheel has radius 0.34 m and her rear sprocket has radius 0.03 m.
A. What is the radius of her front sprocket?
B. Through what angle (in degrees) does her rear wheel rotate in 0.004 second?
To find the radius of her front sprocket, we can use the formula:
Rear Wheel Speed = Front Sprocket Speed * Rear Wheel Radius / Front Sprocket Radius
Given:
Rear Wheel Speed = 27 km/hour = 27,000 m/3600 sec = 7.5 m/sec
Rear Wheel Radius = 0.34 m
Rear Sprocket Radius = 0.03 m
Let's assume the Front Sprocket Radius = x.
Using the formula, we can write:
7.5 m/sec = 95 rpm * 0.34 m / x
Rearranging the equation, we get:
x = (95 rpm * 0.34 m) / 7.5 m/sec
x = 4.3 rpm
Therefore, the radius of her front sprocket is 4.3 rpm.
To find the angle through which her rear wheel rotates in 0.004 seconds, we can use the formula:
Angle (in radians) = Angular Speed (in rad/s) * Time (in seconds)
Given:
Rear Wheel Speed = 95 rpm
Time = 0.004 seconds
Let's convert the rear wheel speed to angular speed:
Angular Speed Rear Wheel = 95 rpm * 2π radians/60 seconds
Angular Speed Rear Wheel = 9.95 rad/s
Now, we can calculate the angle:
Angle (in radians) = 9.95 rad/s * 0.004 seconds
Angle (in radians) = 0.0398 radians
To convert radians to degrees, we multiply by 180/π:
Angle (in degrees) = 0.0398 radians * 180/π degrees
Angle (in degrees) ≈ 2.28 degrees
Therefore, the rear wheel rotates through approximately 2.28 degrees in 0.004 seconds.
To find the radius of her front sprocket, we can use the gear ratio formula:
Gear Ratio = Radius of Rear Sprocket / Radius of Front Sprocket
Given:
Radius of Rear Sprocket = 0.03 m
Let's denote the radius of the front sprocket as R.
From the gear ratio formula, we have:
0.03 m / R = 27 km/h / 95 rpm
To convert 27 km/h to m/s, we divide by 3.6 since 1 km/h = 1000 m/3600 s:
27 km/h = 27,000 m/3600 s = 7.5 m/s
Similarly, to convert 95 rpm to radians per second, we multiply by 2π/60 since there are 2π radians in a full revolution and 60 seconds in a minute:
95 rpm = 95 * 2π/60 radians/second ≈ 9.95 radians/second
Substituting the values in the equation:
0.03 m / R = 7.5 m/s / 9.95 rad/s
Cross-multiplying:
0.03 m * 9.95 rad/s = R * 7.5 m/s
Simplifying:
0.2985 m²/s = 7.5 R m/s
Dividing both sides by 7.5 m/s:
R = 0.2985 m²/s / 7.5 m/s
Simplifying:
R ≈ 0.0398 m
Therefore, the radius of her front sprocket is approximately 0.0398 m or 3.98 cm.
For part B, we can calculate the angular displacement of the rear wheel in 0.004 seconds.
Given:
Rear wheel radius = 0.34 m
The linear speed of the rear wheel is given by the formula:
Linear Speed = Angular Speed * Radius
To find the angular speed (ω) in radians per second:
Linear Speed = 27 km/h = 7.5 m/s (from earlier calculation)
Angular Speed = Linear Speed / Rear Wheel Radius
Substituting the values:
Angular Speed = 7.5 m/s / 0.34 m
Angular Speed ≈ 22.0588 rad/s
To find the angular displacement (θ) in radians, we can use the formula:
θ = Angular Speed * Time
Substituting the values:
θ = 22.0588 rad/s * 0.004 s
θ ≈ 0.08823 radians
To convert the angular displacement from radians to degrees, we multiply by the conversion factor:
0.08823 radians * (180° / π radians)
θ ≈ 5.058 degrees
Therefore, the rear wheel rotates approximately 5.058 degrees in 0.004 seconds.