Question

A circular wheel 28 inches in diameter rotates the same number of inches per second as a circular wheel 35 inches diameter. If the smaller wheel makes x revolutions per second, how many revolutions per minute does the larger wheel make in terms of x.

Answer 1

One revolution of a circle is the circumference of that circle. First find out the circumference of the wheel with the diameter of 28 inches, to get x revolutions per second multiply with x, convert x revolution per second to x revolutions per minute. Assuming, meanwhile, the wheel with 35 inches of diameter makes 'n' revolutions per minute. Equate the revolutions per minute of both the wheels.

Complete step-by-step answer :
We know that,
1 revolution of a circle = circumference of that circle.
Here, the circle is our wheel, which rotates the same number of inches per second.
Since, circumference is the revolution of wheel, so,
1 revolution of a circle/ wheel with the diameter of 28 inches would be $\Rightarrow \pi d$

& \text{Circumference of a circle = 2}\times \pi \times \text{radius} \\\ & \Rightarrow \text{2}\times \text{r}\times \pi \\\ \end{aligned}$$ We know that the term 2r is nothing but diameter $$\Rightarrow d\times \pi $$ Now, we can equate the above information as $$\pi d=28\pi \text{ inches}$$ Hence, x revolution per second $$28\pi .x\text{ inches per second}$$ We know 1 minute = 60 seconds. So, revolution $$28\pi \times x\times 60\text{ inches per minute}$$ Suppose, the wheel with 35 inches diameter rotates 'n' inches per minute while the wheel with 28 inches diameter makes 'x' revolutions per minute. Therefore, n revolution of circle/ wheel with diameter 35 inches $$\Rightarrow 35\times \pi \times n$$ Given that, the revolutions made by both the wheels are equal, that is, $$\begin{aligned} & 35\pi n=28\pi x\times 60 \\\ & \Rightarrow n=\dfrac{28\pi x\times 60}{35\pi } \\\ \end{aligned}$$ Therefore, n = 48x inches per minute. Revolution made per minute by wheel with 35 inches diameter is 48x. **Note** :Do not get confused with the radius and diameter of the circle while putting the values in the circumference formula. Students need to convert the unit inches in meter or centimeter here. Here the students must remember that revolution per minute is nothing but speed in units/minute divided by the circumference in the same unit. So therefore, as long as the unit of speed and the circumference is the same, there is no need for unit conversion here.