J and K are cycling together. K is riding a bike with a wheel radius of 35cm, crank
wheel radius 11cm and chain wheel radius 4cm. She is pedalling at 60 rpm. How
fast is she going?
-->What is the significance of giving 3 different types of radius?
--> know to find circumference and multiply it by rpm to find linear velocity
--> but what to radius to use/how?
The three different types of radius are relevant because the crank wheel radius, chain wheel radius, and wheel radius each contribute to the overall movement of the bike.
First, we need to find the gear ratio, which is the ratio between the number of teeth on the crank wheel (front gear) and the number of teeth on the chain wheel (rear gear). The gear ratio can be calculated using the radii of the crank wheel and chain wheel:
Gear ratio = (Crank wheel radius) / (Chain wheel radius) = 11cm / 4cm = 2.75
Next, we need to find the effective wheel rotation per minute due to this gear ratio. To do this, multiply the pedal rotation per minute by the gear ratio:
Effective wheel rotations per minute = (Pedaling rpm) × (Gear ratio) = 60 rpm × 2.75 = 165 rpm
Now, we can calculate the distance traveled per minute by multiplying the wheel's circumference by the effective wheel rotations per minute. The wheel's circumference can be found using its radius:
Circumference = 2 × π × wheel radius = 2 × π × 35cm = 220 cm (approx.)
Distance traveled per minute = (Wheel circumference) × (Effective wheel rotations per minute) = 220 cm × 165 rpm = 36300 cm/minute
Finally, we can convert the distance traveled per minute into a more standard unit, such as kilometers per hour:
36300 cm/minute * (1 m/100 cm) * (1 km/1000 m) * (60 minutes/1 hour) = 21.78 km/h
So, K is going at a speed of approximately 21.78 km/h.
The three different radii provided in the question (wheel radius, crank wheel radius, and chain wheel radius) are significant for calculating the linear velocity of K while cycling. Each radius represents a different part of the bike's gear system and affects the overall speed.
To calculate the linear velocity, we need to use the effective radius of the gear system. The effective radius is calculated by considering the combination of the chain wheel radius, crank wheel radius, and wheel radius.
Let's calculate the effective radius:
1. First, let's find the effective radius at the chain wheel:
Effective radius at the chain wheel = chain wheel radius = 4 cm
2. Next, let's find the effective radius at the crank wheel:
Effective radius at the crank wheel = crank wheel radius + chain wheel radius
Effective radius at the crank wheel = 11 cm + 4 cm = 15 cm
3. Finally, let's find the effective radius at the wheel:
Effective radius at the wheel = wheel radius + crank wheel radius
Effective radius at the wheel = 35 cm + 15 cm = 50 cm
Now that we have calculated the effective radius, we can proceed to calculate K's speed using linear velocity.
To find the linear velocity:
4. Calculate the circumference of the effective gear system:
Circumference = 2 * pi * Effective radius
Circumference = 2 * pi * 50 cm
5. Convert K's pedaling rate from rpm to revolutions per second:
Pedaling rate in revolutions per second = 60 rpm / 60 seconds
Pedaling rate in revolutions per second = 1 revolution per second
6. Calculate K's linear velocity:
Linear Velocity = Circumference * Pedaling rate in revolutions per second
Now you can plug in the values and calculate the linear velocity of K.
The three different types of radius given in the question (wheel radius, crank wheel radius, and chain wheel radius) are all used to calculate different distances and speeds related to the bike's motion.
1. Wheel Radius: The wheel radius refers to the radius of the bike's main wheel. In this case, the wheel radius is given as 35cm. This radius is used to find the circumference of the wheel, which in turn helps determine the distance covered in one revolution.
2. Crank Wheel Radius: The crank wheel radius refers to the radius of the bike's pedal crank. In this case, the crank wheel radius is given as 11cm. This radius is used to calculate the distance covered by the pedal crank in one revolution.
3. Chain Wheel Radius: The chain wheel radius refers to the radius of the chain wheel, which is attached to the pedal crank. In this case, the chain wheel radius is given as 4cm. This radius is used to determine the distance covered by the chain in one revolution.
To calculate how fast K is going, we need to convert the rotational speed (rpm) into linear velocity (m/s or km/h). Here's how you can do it:
1. Calculate the distance covered by the wheel in one revolution:
Wheel Circumference = 2 x π x Wheel Radius
Distance covered by the wheel in one revolution = Wheel Circumference
2. Calculate the distance covered by the pedal crank in one revolution:
Crank Circumference = 2 x π x Crank Wheel Radius
Distance covered by the pedal crank in one revolution = Crank Circumference
3. Calculate the distance covered by the chain in one revolution:
Chain Circumference = 2 x π x Chain Wheel Radius
Distance covered by the chain in one revolution = Chain Circumference
4. Convert the given pedal speed of 60 rpm into radians per second (ω):
ω = (2 x π x Pedal Speed) / 60
5. Calculate the linear velocity (V) using the formula:
V = Distance covered by the wheel in one revolution x ω
By using these calculations, you can determine the speed at which K is going.