To find the centripetal force $F_c$ that the child must exert to stay on the merry-go-round, we need to use the formula for centripetal force:

$F_c = m \cdot a_c$

where:

• $m$ is the mass of the child,
• $a_c$ is the centripetal acceleration.

Step 1: Convert the Rotational Speed to Radians per Second

The given rotational speed is in revolutions per minute (rpm), so we need to convert it to radians per second.

1. The merry-go-round rotates at $45.0 \, \text{rpm}$.
2. Convert rpm to revolutions per second: $\frac{45.0 \, \text{rpm}}{60} = 0.75 \, \text{revolutions per second}$
3. Convert revolutions per second to radians per second, noting that one revolution is $2\pi$ radians: $\omega = 0.75 \times 2\pi = 1.5\pi \approx 4.71 \, \text{rad/s}$

Step 2: Calculate Centripetal Acceleration $a_c$

The centripetal acceleration is given by:

$a_c = \omega^2 \cdot r$

where:

• $\omega = 4.71 \, \text{rad/s}$ (angular velocity),
• $r = 2.00 \, \text{m}$ (distance from the center).

Substitute the values:

$a_c = (4.71)^2 \times 2.00 \approx 44.34 \, \text{m/s}^2$

Step 3: Calculate Centripetal Force $F_c$

Now, use the mass of the child ($m = 23.0 \, \text{kg}$) and the centripetal acceleration:

$F_c = m \cdot a_c = 23.0 \times 44.34 \approx 1019.8 \, \text{N}$

Answer

The centripetal force the child must exert to stay on the merry-go-round is approximately:

$F_c \approx 1019.8 \, \text{N}$

Would you like more details or have any questions?

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Related Questions

1. How would the centripetal force change if the child was closer to the center, say at 1.0 m?
2. What happens to the centripetal force if the merry-go-round's speed increases?
3. How is centripetal acceleration related to the radius and angular velocity?
4. What would be the required force if the child’s mass increased to 30 kg?
5. If the merry-go-round was rotating at 60 rpm, what would the new centripetal force be?

Tip

When dealing with rotational motion, always ensure to convert angular speeds to consistent units, typically radians per second for calculations in physics.