QuestionAugust 2, 2025

2. A playground merry-go -round with a radius of 2.80 meters starts from rest, and is pushed so that it accelerates at a constant rate of -0.192rad/s^2 up until it reaches a maximum angular speed of 0.4rev/s a. How long (in seconds)does it take the merry-go-round to reach this maximum angular speed?[6 pts] b. Through how many radians has the merry-go-round turned during the time interval found in part a? [5 pts] c. In what direction is the merry-go-round spinning?Clockwise or counterclockwise? [2 pts] d. A23 kg kid is standing at the edge of the merry-go -round. What is the magnitude of their centripetal acceleration (inm/s^2) at the moment the merry-go-round reaches its maximum angular speed?[4 pts] e. What is the magnitude of the centripetal force (in N) acting on this kid at the moment the merry-go- round reaches its maximum angular speed? [4 pts] f. What is the direction of the centripetal force acting on this kid at the moment the merry.go-round reaches its maximum angular speed? Circle One. [2 pts] Tangent to the merry -go-round and pointing in the counterclockwise direction Tangent to the merry -go-round and pointing in the clockwise direction Radially inward Radially outward Unknown. Not enough information given. g. What is the magnitude of this kid's linear velocity (in m/s ) at the moment the merry-go-round reaches its maximum angular speed?[4 pts]

2. A playground merry-go -round with a radius of 2.80 meters starts from rest, and is pushed so that it accelerates at a constant rate of -0.192rad/s^2 up until it reaches a maximum angular speed of 0.4rev/s a. How long (in seconds)does it take the merry-go-round to reach this maximum angular speed?[6 pts] b. Through how many radians has the merry-go-round turned during the time interval found in part a? [5 pts] c. In what direction is the merry-go-round spinning?Clockwise or counterclockwise? [2 pts] d. A23 kg kid is standing at the edge of the merry-go -round. What is the magnitude of their centripetal acceleration (inm/s^2) at the moment the merry-go-round reaches its maximum angular speed?[4 pts] e. What is the magnitude of the centripetal force (in N) acting on this kid at the moment the merry-go- round reaches its maximum angular speed? [4 pts] f. What is the direction of the centripetal force acting on this kid at the moment the merry.go-round reaches its maximum angular speed? Circle One. [2 pts] Tangent to the merry -go-round and pointing in the counterclockwise direction Tangent to the merry -go-round and pointing in the clockwise direction Radially inward Radially outward Unknown. Not enough information given. g. What is the magnitude of this kid's linear velocity (in m/s ) at the moment the merry-go-round reaches its maximum angular speed?[4 pts]
2. A playground merry-go -round with a radius of 2.80 meters starts from rest, and is pushed so that it accelerates at a constant rate of -0.192rad/s^2 up until it reaches a maximum angular speed of 0.4rev/s a. How long (in seconds)does it take the merry-go-round to reach this maximum angular speed?[6 pts] b. Through how many radians has the merry-go-round turned during the time interval found in part a? [5 pts] c. In what direction is the merry-go-round spinning?Clockwise or counterclockwise? [2 pts] d. A23 kg kid is standing at the edge of the merry-go -round. What is the magnitude of their centripetal acceleration (inm/s^2) at the moment the merry-go-round reaches its maximum angular speed?[4 pts] e. What is the magnitude of the centripetal force (in N) acting on this kid at the moment the merry-go- round reaches its maximum angular speed? [4 pts] f. What is the direction of the centripetal force acting on this kid at the moment the merry.go-round reaches its maximum angular speed? Circle One. [2 pts] Tangent to the merry -go-round and pointing in the counterclockwise direction Tangent to the merry -go-round and pointing in the clockwise direction Radially inward Radially outward Unknown. Not enough information given. g. What is the magnitude of this kid's linear velocity (in m/s ) at the moment the merry-go-round reaches its maximum angular speed?[4 pts]

Solution
4.6(290 votes)

Answer

a. t = \frac{0.8\pi}{0.192} \approx 13.09 \text{ s} ### b. \theta = \frac{1}{2}(-0.192)(13.09)^2 \approx -16.36 \text{ rad} ### c. Clockwise ### d. a_c = 2.80 \times (0.8\pi)^2 \approx 17.67 \text{ m/s}^2 ### e. F_c = 23 \times 17.67 \approx 406.41 \text{ N} ### f. Radially inward ### g. v = 2.80 \times 0.8\pi \approx 7.04 \text{ m/s} Explanation 1. Convert angular speed to rad/s 0.4 \text{ rev/s} = 0.4 \times 2\pi \text{ rad/s} = 0.8\pi \text{ rad/s} 2. Calculate time to reach maximum angular speed Use \omega = \omega_0 + \alpha t. Here, \omega_0 = 0, \alpha = -0.192 \text{ rad/s}^2, \omega = 0.8\pi \text{ rad/s}. Solve for t: t = \frac{\omega}{\alpha} = \frac{0.8\pi}{0.192}. 3. Calculate radians turned during acceleration Use \theta = \omega_0 t + \frac{1}{2}\alpha t^2. Here, \omega_0 = 0, solve for \theta: \theta = \frac{1}{2}(-0.192)t^2. 4. Determine spinning direction Negative acceleration implies clockwise direction. 5. Calculate centripetal acceleration Use a_c = r\omega^2. Here, r = 2.80 \text{ m}, \omega = 0.8\pi \text{ rad/s}. 6. Calculate centripetal force Use F_c = ma_c. Here, m = 23 \text{ kg}. 7. Determine direction of centripetal force Centripetal force is always radially inward. 8. Calculate linear velocity Use v = r\omega. Here, r = 2.80 \text{ m}, \omega = 0.8\pi \text{ rad/s}.

Explanation

1. Convert angular speed to rad/s<br /> $0.4 \text{ rev/s} = 0.4 \times 2\pi \text{ rad/s} = 0.8\pi \text{ rad/s}$<br /><br />2. Calculate time to reach maximum angular speed<br /> Use $\omega = \omega_0 + \alpha t$. Here, $\omega_0 = 0$, $\alpha = -0.192 \text{ rad/s}^2$, $\omega = 0.8\pi \text{ rad/s}$. Solve for $t$: $t = \frac{\omega}{\alpha} = \frac{0.8\pi}{0.192}$.<br /><br />3. Calculate radians turned during acceleration<br /> Use $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$. Here, $\omega_0 = 0$, solve for $\theta$: $\theta = \frac{1}{2}(-0.192)t^2$.<br /><br />4. Determine spinning direction<br /> Negative acceleration implies clockwise direction.<br /><br />5. Calculate centripetal acceleration<br /> Use $a_c = r\omega^2$. Here, $r = 2.80 \text{ m}$, $\omega = 0.8\pi \text{ rad/s}$.<br /><br />6. Calculate centripetal force<br /> Use $F_c = ma_c$. Here, $m = 23 \text{ kg}$.<br /><br />7. Determine direction of centripetal force<br /> Centripetal force is always radially inward.<br /><br />8. Calculate linear velocity<br /> Use $v = r\omega$. Here, $r = 2.80 \text{ m}$, $\omega = 0.8\pi \text{ rad/s}$.
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