the front wheel of a bicycle has a diameter of 16 in and the back wheel has a diameter of 26 in. through what angles does the front wheel turn if the rear wheel turns through 24 radians?
The product of radius and angle turned through is the same for both wheels. The smaller wheel turns faster.
24*26 = 16*theta
theta = 1.5*26 = 39 radians
To find out the angles through which the front wheel turns, we can use the relationship between the sizes of the wheels and the distances they cover in a rotation. The distance covered by a wheel in one complete rotation is equal to its circumference.
Let's calculate the circumference of the front and back wheels first. The formula to find the circumference of a circle is given by:
Circumference = π * Diameter
For the front wheel:
C1 = π * 16 in
For the back wheel:
C2 = π * 26 in
Now, we need to determine how many complete rotations the rear wheel makes. We are given that the rear wheel turns through 24 radians. To find the number of rotations, we use the following relationship:
Number of rotations = Radians / (2 * π)
Number of rotations = 24 radians / (2 * π)
Once we have the number of rotations, we can calculate the distance covered by the rear wheel during this rotation. Since the circumference of the rear wheel represents the distance covered in one rotation, the distance can be found by multiplying the number of rotations by the circumference:
Distance covered by the rear wheel = Number of rotations * C2
To find out the angle through which the front wheel turns, we need to determine the ratio between the distances covered by the front and rear wheels. This ratio is equal to the ratio of their respective circumferences:
Distance covered by the front wheel / Distance covered by the rear wheel = C1 / C2
Now, we can substitute the values we calculated earlier into the equation:
Angle of rotation for the front wheel = (Distance covered by the front wheel / Distance covered by the rear wheel) * 24 radians
Substituting the values, we get:
Angle of rotation for the front wheel = (C1 / C2) * 24 radians
Now, let's plug in the values we obtained:
Angle of rotation for the front wheel = (π * 16 in) / (π * 26 in) * 24 radians
Simplifying further:
Angle of rotation for the front wheel = (16/26) * 24 radians
And finally:
Angle of rotation for the front wheel ≈ 14.769 radians