a trail bike has a front wheel with a diameter of 40cm and a back wheel of of a diametet 60cm. Through what angle does the front wheel turn if the back wheel turns through 8 radians?
Why did the front wheel join a yoga class? Because it wanted to find its inner angle!
Alright, let's get serious now. To find the angle through which the front wheel turns when the back wheel turns through 8 radians, we can use the concept of angular displacement.
The angular displacement of a wheel can be calculated using the formula:
θ = s / r
Where θ is the angular displacement, s is the linear displacement along the circumference, and r is the radius of the wheel.
Now, the linear displacement of the back wheel can be found using its circumference:
s_back = 2π * r_back
= 2π * (diameter_back / 2)
= π * diameter_back
Substituting the value of the back wheel's diameter (60cm):
s_back = π * 60cm
= 60π cm
Similarly, the linear displacement of the front wheel can be found using its circumference:
s_front = π * diameter_front
= π * 40cm
Now, we can find the angular displacement of the front wheel:
θ_front = s_front / r_front
= (π * 40cm) / (20cm)
= 2π radians
So, the front wheel turns through an angle of 2π radians when the back wheel turns through 8 radians.
But hey, don't be too concerned, the front wheel is just rotating to keep up with the back wheel's wild moves on the trail!
To calculate the angle through which the front wheel turns, we need to determine the relative difference in circumference between the front and back wheels.
The circumference of a circle can be calculated using the formula: C = 2πr, where C is the circumference and r is the radius.
For the front wheel (diameter of 40cm), the radius is half of the diameter, so the radius is 20cm (0.2m).
The circumference of the front wheel is therefore: C_front = 2π * 0.2 = 1.26m.
For the back wheel (diameter of 60cm), the radius is half of the diameter, so the radius is 30cm (0.3m).
The circumference of the back wheel is therefore: C_back = 2π * 0.3 = 1.88m.
Now, we need to find the ratio of the front wheel's circumference to the back wheel's circumference:
Ratio = C_front / C_back = 1.26 / 1.88.
If the back wheel turns through 8 radians, the front wheel will also turn through the same ratio:
Angle_front = Ratio * 8 radians.
Substituting the values, we get:
Angle_front = (1.26 / 1.88) * 8 radians.
Now we can calculate the angle step-by-step:
Step 1: Simplify the ratio: 1.26 / 1.88 = 0.67
Step 2: Calculate the final angle:
Angle_front = 0.67 * 8 radians = 5.36 radians.
Therefore, the front wheel turns through approximately 5.36 radians.
To find the angle through which the front wheel turns, we need to compare the distances covered by the front and back wheels. The distance covered by a wheel can be calculated by multiplying its circumference with the number of revolutions it completes.
The circumference of a circle can be calculated using the formula C = πd, where C is the circumference and d is the diameter.
For the front wheel:
C_front = π * 40 cm = 40π cm
For the back wheel:
C_back = π * 60 cm = 60π cm
Since the back wheel turns through 8 radians, we can use the relationship between radians, circumference, and distance to find the distance covered by the back wheel.
Distance_back = C_back * 8 = 60π * 8 = 480π cm
Now, to find the angle through which the front wheel turns, we need to calculate the number of revolutions it completes. Since the front and back wheels must cover the same distance:
Distance_front = C_front * number_of_revolutions
We can rearrange the equation to solve for the number of revolutions:
number_of_revolutions = Distance_front / C_front
= Distance_back / C_front
= (480π cm) / (40π cm)
= 12 revolutions
Since one revolution is equivalent to 2π radians, the angle through which the front wheel turns is:
Angle_front = number_of_revolutions * 2π
= 12 * 2π
= 24π radians
Therefore, the front wheel turns through an angle of 24π radians when the back wheel turns through 8 radians.
since the wheels travel the same distance, and s = rθ,
40θ = 60*8
or, since the front wheel is 3/2 as big, it turns 3/2 as fast.