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During a certain period of time, the angular displacement of a swinging door is described by 𝜃 = 5.00 + 10.0𝑡 + 2.00𝑡2 where θ is radians and t is in seconds. Determine the angular displacement, angular speed and angular acceleration, (a) at time t = 0 s, (b) at time t = 3.00 s.
alphonsio

• At $t = 0$ s:

• Angular Displacement: $\theta = 5$ radians
• Angular Speed: $\omega = 10$ rad/s
• Angular Acceleration: $\alpha = 4$ rad/s²
• At $t = 3$ s:

• Angular Displacement: $\theta = 53$ radians
• Angular Speed: $\omega = 22$ rad/s
• Angular Acceleration: $\alpha = 4$ rad/s²

Given the equation for the angular displacement $\theta(t)$ as a function of time $t$:

$\theta(t) = 5 + 10t + 2t^2$

where $\theta$ is in radians and $t$ is in seconds. We need to determine the angular displacement, angular speed, and angular acceleration at times $t = 0$ s and $t = 3.00$ s.

1. Angular Displacement, $\theta(t)$

This is given directly by the equation :

$\theta(t) = 5 + 10t + 2t^2$

2. Angular Speed, $\omega(t)$

The angular speed is the first derivative of the angular displacement with respect to time:

$\omega(t) = \frac{d\theta(t)}{dt}$

Let's differentiate the given equation:

$\omega(t) = \frac{d}{dt} \left(5 + 10t + 2t^2\right) = 0 + 10 + 4t$

So,

$\omega(t) = 10 + 4t \quad \text{(in radians per second)}$

3. Angular Acceleration, $\alpha(t)$

The angular acceleration is the second derivative of the angular displacement with respect to time, or the first derivative of the angular speed with respect to time:

$\alpha(t) = \frac{d\omega(t)}{dt}$

Let's differentiate $\omega(t)$ :

$\alpha(t) = \frac{d}{dt} (10 + 4t) = 0 + 4 = 4 \quad \text{(in radians per second squared)}$

So the angular acceleration $\alpha(t)$ is constant and equal to 4 radians per second squared.

Now, let's calculate the values at $t = 0$ s and $t = 3$ s.

(a) At $t = 0$ s:

1. Angular Displacement $\theta(0)$:

$\theta(0) = 5 + 10(0) + 2(0)^2 = 5 \text{ radians}$

1. Angular Speed $\omega(0)$:

$\omega(0) = 10 + 4(0) = 10 \text{ radians per second}$

1. Angular Acceleration $\alpha(0)$:

$\alpha(0) = 4 \text{ radians per second squared}$

(b) At $t = 3$ s:

1. Angular Displacement $\theta(3)$ :

$\theta(3) = 5 + 10(3) + 2(3)^2 = 5 + 30 + 18 = 53 \text{ radians}$

1. Angular Speed $\omega(3.00)$ :

$\omega(3) = 10 + 4(3) = 10 + 12 = 22 \text{ radians per second}$

1. Angular Acceleration $\alpha(3)$:

$\alpha(3) = 4 \text{ radians per second squared}$

Summary:

• At $t = 0$ s:

• Angular Displacement: $\theta = 5$ radians
• Angular Speed: $\omega = 10$ rad/s
• Angular Acceleration: $\alpha = 4$ rad/s²
• At $t = 3$ s:

• Angular Displacement: $\theta = 53$ radians
• Angular Speed: $\omega = 22$ rad/s
• Angular Acceleration: $\alpha = 4$ rad/s²