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

Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problem 43

Chapter 4, Problem 43

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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Step 1: Convert the angular speed from revolutions per minute (rpm) to radians per second (rad/s). Use the formula: $ \omega = \frac{\text{rpm} \times 2\pi}{60} $. Substitute $ \text{rpm} = 500 $ to find $ \omega $ in rad/s.

Step 2: Calculate the angular acceleration during the first phase (acceleration phase). Use the formula: $ \alpha = \frac{\Delta \omega}{\Delta t} $, where $ \Delta \omega $ is the change in angular velocity and $ \Delta t $ is the time interval (1.0 s).

Step 3: Determine the angular displacement during the acceleration phase using the kinematic equation: $ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $, where $ \omega_0 = 0 $ (starting from rest), $ \alpha $ is the angular acceleration, and $ t = 1.0 $ s.

Step 4: Calculate the angular displacement during the constant angular velocity phase. Use the formula: $ \theta = \omega t $, where $ \omega $ is the constant angular velocity (calculated in Step 1) and $ t = 3.0 $ s.

Step 5: Calculate the angular displacement during the deceleration phase. First, find the angular deceleration $ \alpha $ using $ \alpha = \frac{\Delta \omega}{\Delta t} $, where $ \Delta \omega $ is the change in angular velocity (from $ \omega $ to 0) and $ \Delta t = 2.0 $ s. Then, use the kinematic equation $ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $, where $ \omega_0 $ is the initial angular velocity (calculated in Step 1). Finally, sum up the angular displacements from all three phases and convert the total angular displacement from radians to revolutions using $ \text{revolutions} = \frac{\theta}{2\pi} $.

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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 the rate of change of angular displacement and is typically measured in radians per second or revolutions per minute (rpm). In this question, the DVD accelerates to an angular velocity of 500 rpm, which is crucial for calculating the total revolutions made during its motion.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time. It is essential for determining how quickly the DVD speeds up from rest to its maximum speed and then slows down to a stop. The acceleration phase lasts 1.0 s, while the deceleration phase lasts 2.0 s, both of which are key to calculating the total revolutions.

Kinematic Equations for Rotational Motion

Kinematic equations for rotational motion are analogous to linear motion equations but apply to angular quantities. These equations allow us to relate angular displacement, angular velocity, and angular acceleration. They are used to calculate the total number of revolutions made by the DVD during its acceleration, constant speed, and deceleration phases.

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Equations of Rotational Motion

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