Question

A typical laboratory centrifuge rotates at 3300 rpm . Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. What is the acceleration at the end of a test tube that is 10 cm from the axis of rotation? Express your answer with the appropriate units. a = 1.2×10⁴ ms²

Answer 1

The acceleration at the end of a test tube that is 10 cm from the axis of rotation is
`\(1.2 * 10^4 \, \text{m/s}^2\).`

Step-by-step explanation:

The centripetal acceleration
`(\(a_c\))` of an object moving in a circular path is given by the formula
`\(a_c = \frac{{v^2}}{{r}}\)`, where v is the linear velocity and r is the radius of the circular path. In this case, the linear velocity is determined by the rotational speed in revolutions per minute (rpm), which needs to be converted to meters per second (m/s). The formula for linear velocity v is
`\(v = \frac{{2\pi r * \text{{rpm}}}}{{60}}\)`. Substituting this into the centripetal acceleration formula and solving for
`\(a_c\)`, we get
`\(a_c = \frac{{4\pi^2 r * \text{{rpm}}^2}}{{3600}}\).`

Given the rotational speed of the centrifuge
`(\(\text{{rpm}} = 3300\))`and the radius
`(\(r = 0.1 \, \text{m}\) for 10 cm)`, we can substitute these values into the formula to calculate
`\(a_c\).` The result is
`\(1.2 * 10^4 \, \text{m/s}^2\),`which represents the acceleration at the end of the test tube. This large acceleration emphasizes the need for careful handling when placing test tubes into a laboratory centrifuge to prevent any undesirable consequences due to the high centrifugal forces generated during rotation.