If an ant starts at 0 radians and travels to 4.5 radians in 18 seconds, what is its angular velocity? If the ant then slows to 0 rad/s in 3 seconds, what is its angular acceleration?
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To calculate the angular velocity, we use the formula:
\[ \text{Angular velocity} (\omega) = \frac{\text{change in angle} (\Delta \theta)}{\text{time taken} (\Delta t)} \]
Given: - The change in angle (∆θ) is 4.5 radians. - The time taken (∆t) is 18 seconds.
Plug in the values:
\[ \omega = \frac{4.5 \text{ radians}}{18 \text{ seconds}} \]
\[ \omega = \frac{1}{4} \text{ radian per second} \]
\[ \omega = 0.25 \text{ rad/s} \]
So, the ant's angular velocity is 0.25 rad/s.
Next, we calculate the angular acceleration. Angular acceleration (α) is the rate of change of angular velocity over time, and its formula is:
\[ \text{Angular acceleration} (\alpha) = \frac{\text{change in angular velocity} (\Delta \omega)}{\text{time taken} (\Delta t)} \]
Given: - The ant slows down to 0 rad/s, hence the change in angular velocity (∆ω) is -0.25 rad/s (since it is slowing down, it is a decrease from the initial angular velocity). - The time taken (∆t) for this change is 3 seconds.
Plug in the values:
\[ \alpha = \frac{-0.25 \text{ rad/s}}{3 \text{ seconds}} \]
\[ \alpha \approx -0.0833 \text{ rad/s}^2 \]
So, the ant's angular acceleration is approximately -0.0833 rad/s².