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A merry-go-round with a radius of 1.25 m at the park is rotating at 75 rpm. Determine the magnitude of the centripetal acceleration.
a. 31.5 m/s
b. 61.7 m/s
c. 77.1 m/s
d. 15.0 m/s

Verified Answer :

jacob193

Answer:

Approximately \(77.1\; {\rm m\cdot s^{-2}}\).

Explanation:

If an object is in a circular motion of radius \(r\) at an angular velocity of \(\omega\), the centripetal acceleration of the object would be:

\(a = \omega^{2} \, r\).

In this question, the angular velocity of the object is given in the unit \(\text{rpm}\), or "revolutions per minute". Apply unit conversion and ensure that angular velocity is measured in the standard unit of (radians) per second:

  • \(1\; {\text{minute}} = 60\; {\rm s}\).
  • \(1\; \text{revolution} = 2\, \pi\) (radians.)

Hence:

\(\begin{aligned}\omega &= 75\; \text{rpm} \\ &= 75\; {\rm (\text{revolution}) \, (\text{minute})^{-1}} \times \frac{1\; \text{minute}}{60\; {\rm s}}\times \frac{2\,\pi}{1\; \text{revolution}} \\ &= \frac{5\, \pi}{2}\; {\rm s^{-1}}\end{aligned}\).

The centripetal acceleration of this object would be:

\(\begin{aligned} a &= \omega^{2} \, r \\ &\approx \left(\frac{5\, \pi}{2}\; {\rm s^{-1}\right)^{2} \, (1.25\; {\rm m}) \\ & \approx 77.1\; {\rm m\cdot s^{-2}}\end{aligned}\).

The correct option c. 77.1 m/s. is The magnitude of the centripetal acceleration is 77.1 m/s.

To determine the magnitude of the centripetal acceleration of a merry-go-round, we need to use the formula:

ac = rω²

where ac is the centripetal acceleration, r is the radius, and ω is the angular velocity in radians per second.

Step-by-step Solution:

First, convert the rotation speed from revolutions per minute (rpm) to radians per second (rad/s). The merry-go-round rotates at 75 rpm.

Convert rpm to radians per second:

1 rpm = 2π radians / 60 seconds

ω = 75 rpm × (2π / 60) = 7.85 rad/s

Next, use the radius of the merry-go-round (r = 1.25 m) and the angular velocity in the centripetal acceleration formula:

Calculate the centripetal acceleration:

ac = rω²

ac = 1.25 m × (7.85 rad/s)²

ac ≈ 77.1 m/s²

The correct answer is option c. 77.1 m/s².

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