Question

A gear is driven by a chain that travels 1.46 meters per second. Find the radius of the gear if it makes 46 revolution per minute.

Answer 1

Using the relationship between linear speed, angular speed, and radius, the gear's radius is found by dividing the linear speed by the angular speed, calculated from the given number of revolutions.

To determine the radius of the gear, we can use the relationship between linear speed, angular speed, and radius. The linear speed v of a point on the edge of a rotating object is given by the product of its angular speed
`(\(\omega\))` and radius (r):

`\[ v = \omega \cdot r \]`

Given that the gear makes 46 revolutions per minute, we first need to find the angular speed
`(\(\omega\))`. One revolution corresponds to
`\(2\pi\)` radians, so the angular speed can be calculated as follows:

`\[ \text{Angular speed} (\omega) = \frac{\text{Number of revolutions}}{\text{Time}} \]`

`\[ \omega = \frac{46 \, \text{revolutions/minute}}{1 \, \text{minute}} * \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \]`

Now, we know that
`\(v = 1.46 \, \text{m/s}\)` and
`\(\omega\)`, and we can rearrange the linear speed formula to solve for the radius:

`\[ r = (v)/(\omega) \]`

Substitute the values into the formula:

`\[ r = \frac{1.46 \, \text{m/s}}{\omega} \]`

After calculating
`\(\omega\)`, we can determine the radius of the gear.