The angle θthrough which a disk drive turns is given by θ(t)=a+b t-c t^3, where a, b, and c are

step 1 Hi, everybody. So we need to find the A, B, and C including their units. Okay. And so for this,

The angle θthrough which a disk drive turns is given by...

The angle $\theta$ through which a disk drive turns is given by $\theta(t)=a+b t-c t^{3},$ where $a, b,$ and $c$ are constants, $t$ is in seconds, and $\theta$ is in radians. When $t=0, \theta=\pi / 4$ rad and the angular velocity is 2.00 $\mathrm{rad} / \mathrm{s}$ , and when $t=1.50 \mathrm{s},$ the angular acceleration is 1.25 $\mathrm{rad} / \mathrm{s}^{2}$ . (a) Find $a, b,$ and $c,$ including their units. (b) What is the angular acceleration when $\theta=\pi / 4$ rad? (c) What are $\theta$ and the angular velocity when the angular acceleration is 3.50 $\mathrm{rad} / \mathrm{s}^{2} ?$

The angle $\theta$ through which a disk drive turns is given by $\theta(t)=a+b t-c t^{3},$ where $a, b,$ and $c$ are constants, $t$ is in seconds, and $\theta$ is in radians. When $t=0, \theta=\pi / 4$ rad and the angular velocity is 2.00 $\mathrm{rad} / \mathrm{s}$ , and when $t=1.50 \mathrm{s},$ the angular
acceleration is 1.25 $\mathrm{rad} / \mathrm{s}^{2}$ . (a) Find $a, b,$ and $c,$ including their units.
(b) What is the angular acceleration when $\theta=\pi / 4$ rad? (c) What are $\theta$ and the angular velocity when the angular acceleration is 3.50 $\mathrm{rad} / \mathrm{s}^{2} ?$

Key Concepts

Solving Equations in Rotational Motion

Problems in rotational dynamics often require solving equations—sometimes polynomial in nature—to determine the time, displacement, or rate of change when a particular condition is met (e.g., a specific angular acceleration). This involves setting up the equations based on kinematic relationships and solving for the unknown variables.

Using Initial Conditions to Determine Constants

Many kinematic problems involve arbitrary constants that arise from integration. These constants are determined by applying initial conditions—known values of displacement, velocity, or acceleration at a specific time. This method ensures that the general solution fits the specific scenario described.

Consistent Unit Analysis

Ensuring that all quantities are expressed in consistent units (such as radians, seconds, radians per second, and radians per second squared) is critical. Proper unit analysis helps confirm that calculations and final answers are dimensionally consistent and physically meaningful.

Angular Kinematics

Angular kinematics describes the rotational motion of objects using quantities such as angular displacement, angular velocity, and angular acceleration. In the problem, these quantities are expressed as functions of time and are interconnected by differentiation, similar to linear kinematics.

Derivatives in Kinematics

To obtain angular velocity and angular acceleration from the angular displacement function, differentiation is used. The first derivative of the displacement function with respect to time gives the angular velocity, while the second derivative yields the angular acceleration. This application of calculus is a key concept in understanding rotational motion.

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