91Ó°ÊÓ

Rotational velocity refers to how fast an object rotates around a fixed axis. In this exercise, the astronaut's rotational velocity in the centrifuge is given by the equation: \ \( \omega(t) = 0.60 \frac{\mathrm{rad}}{\mathrm{s}^2} t \). \ This equation shows that the rotational velocity increases linearly with time. At \(t = 5.0 \mathrm{s}\), substituting in the equation gives: \ \( \omega = 0.60 \times 5.0 = 3.0 \frac{\mathrm{rad}}{\mathrm{s}} \). \ This means the centrifuge spins at \(3.0 \frac{\mathrm{rad}}{\mathrm{s}} \) after 5 seconds. \( \mathrm{Radians}/ \mathrm{second} \) is the unit of measurement here, showing the angle covered per second in radians. Rotational velocity is key in understanding the overall rotational movement of an object around a point.

Translational velocity is the speed at which an object moves through space in a straight line. It is closely linked to rotational velocity by the equation \(v = r \omega\), where \(v\) represents translational velocity, \(r\) is the radius, and \(\omega\) is rotational velocity.
Given that the radius \(r = 10 \mathrm{~m}\) and \(\omega = 3 \frac{\mathrm{rad}}{\mathrm{s}} \), the astronaut's translational velocity can be calculated as: \ \( v = 10 \mathrm{~m} \times 3 \frac{\mathrm{rad}}{\mathrm{s}} = 30 \frac{\mathrm{m}}{\mathrm{s}} \).
This indicates that the astronaut is moving at \(30 \mathrm{m}/ \mathrm{s} \) along the circular path at \(t = 5 \mathrm{~s}\). Translational velocity helps to understand how an orbiting object can translate its circular motion into linear speed if it were to move in a straight line.

Tangential acceleration is the rate of change of translational (linear) velocity along a path tangent to the curve at the point of interest. This can be found by differentiating the rotational velocity with respect to time: \ \( a_t = \frac{d \omega}{d t} \). \ Using the given rotational velocity \( \omega = 0.60 \frac{\mathrm{rad}}{\mathrm{s}^2} t \), the differentiation gives: \( a_t = 0.60 \frac{\mathrm{rad}}{\mathrm{s}^2} \). \
Hence, the tangential acceleration remains constant at \( 0.60 \frac{\mathrm{rad}}{\mathrm{s}^2} \), which implies that the linear speed of the astronaut is increasing at that constant rate. Understanding tangential acceleration is essential for analyzing how the speed of an object changes over time while moving along a curved path.

Radial acceleration, also known as centripetal acceleration, is the rate at which an object changes direction as it moves along a circular path. It always points towards the center of the circular path. The formula for radial acceleration is: \ \( a_r = \omega^2 r \). \ Here, \(\omega = 3.0 \frac{\mathrm{rad}}{\mathrm{s}} \) and \( r = 10 \mathrm{~m} \). Substituting these values: \ \( a_r = (3.0 \frac{\mathrm{rad}}{\mathrm{s}})^2 \times 10 \mathrm{~m} = 90 \frac{\mathrm{m}}{\mathrm{s}^2} \). \ This means the astronaut experiences a radial acceleration of \(90 \frac{\mathrm{m}}{\mathrm{s}^2} \) directed towards the centrifuge's center. Radial acceleration is crucial in analyzing forces that keep an object in circular motion, making it central to understanding rotation dynamics.