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Ch 04: Kinematics in Two Dimensions
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 04: Kinematics in Two DimensionsProblem 40b
Chapter 4, Problem 40b

A 5.0-m-diameter merry-go-round is initially turning with a 4.0 s period. It slows down and stops in 20 s. How many revolutions does the merry-go-round make as it stops?

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Step 1: Calculate the initial angular velocity (ω₀) of the merry-go-round using the formula ω₀ = 2π / T, where T is the period of rotation. Here, T = 4.0 s.
Step 2: Determine the angular acceleration (α) using the formula α = (ω - ω₀) / t, where ω is the final angular velocity (0 rad/s since it stops), ω₀ is the initial angular velocity, and t is the time it takes to stop (20 s).
Step 3: Use the kinematic equation for rotational motion θ = ω₀t + (1/2)αt² to calculate the total angular displacement (θ) in radians. Substitute the values of ω₀, α, and t into the equation.
Step 4: Convert the angular displacement θ from radians to revolutions using the relationship 1 revolution = 2π radians. Divide θ by 2π to find the number of revolutions.
Step 5: Summarize the process and ensure all units are consistent throughout the calculations to verify the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second. For a merry-go-round, it can be calculated from the period of rotation, which is the time taken for one complete revolution. The relationship is given by the formula ω = 2π/T, where ω is the angular velocity and T is the period.
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Angular Deceleration

Angular deceleration refers to the rate at which an object's angular velocity decreases over time. It is crucial for understanding how quickly the merry-go-round slows down. This can be calculated by dividing the change in angular velocity by the time taken to stop, allowing us to determine how many revolutions occur during the deceleration phase.
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Total Revolutions

Total revolutions represent the cumulative number of complete turns an object makes during a specific time interval. For the merry-go-round, this can be calculated by integrating the angular velocity over the time it takes to stop. By knowing the initial angular velocity and the angular deceleration, one can find the total number of revolutions made as it comes to a halt.
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Related Practice
Textbook Question

Starting from rest, a DVD steadily accelerates to 500 rpm in 1.0 s, rotates at this angular speed for 3.0 s, then steadily decelerates to a halt in 2.0 s. How many revolutions does it make?

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Textbook Question

FIGURE EX4.36 shows the angular velocity graph of the crankshaft in a car. What is the crankshaft's angular acceleration at t = 3s

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Textbook Question

Your roommate is working on his bicycle and has the bike upside down. He spins the 60-cm-diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. What are the pebble's speed and acceleration?

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Textbook Question

A speck of dust on a spinning DVD has a centripetal acceleration of 20 m/s3 . What would be the acceleration of the first speck of dust if the disk's angular velocity was doubled?

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Textbook Question

A projectile's horizontal range over level ground is v02sin2θg\(\frac{v_0^2 \sin 2\theta}{g}\). At what launch angle or angles will the projectile land at half of its maximum possible range?

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Textbook Question

A spaceship maneuvering near Planet Zeta is located at r=(600i−400j+200k)×103km,\(\mathbf{r}\) = (600\(\mathbf{i}\) - 400\(\mathbf{j}\) + 200\(\mathbf{k}\)) \(\times\) 10^3 \, \(\text{km}\), relative to the planet, and traveling at v=9500im/s\(\mathbf{v}\) = 9500\(\mathbf{i}\) \, \(\text{m/s}\). It turns on its thruster engine and accelerates with a=(40i−20k)m/s2\(\mathbf{a}\) = (40\(\mathbf{i}\) - 20\(\mathbf{k}\)) \, \(\text{m/s}\)^2 for 35 min35\(\text{ min}\). What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in km\(\operatorname{km}\).

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