Problem 20 - Applied Mathematics

Question

A particle is moving in the positive sense on the circumference of a circle of radius \(2 \mathrm{~m}\). The particle has a constant angular acceleration of \(3 \mathrm{rad} \mathrm{s}^{-2}\). At one instant the speed of the particle is \(2 \mathrm{~ms}^{-1}\) and one second later it is \(8 \mathrm{~ms}^{-1}\).

Step-by-Step Solution

Answer

Initial angular velocity is 1 rad/s; final angular velocity is 4 rad/s; angular acceleration is 3 rad/s².

Step 1 - Identify known values

List the given values in the problem:- Radius of the circle, \[r = 2 \text{ m}\]- Angular acceleration, \[\alpha = 3 \text{ rad/s}^2\]- Initial speed, \[v_i = 2 \text{ m/s}\]- Speed after 1 second, \[v_f = 8 \text{ m/s}\]- Time, \[t = 1 \text{ s}\]

Step 2 - Relate linear and angular quantities

Use the relationship between linear velocity \(v\) and angular velocity \(\omega\):\[ v = r \omega\]Thus,\[ v_i = r \omega_i = 2 \omega_i \implies \omega_i = \frac{v_i}{r} = \frac{2 \text{ m/s}}{2 \text{ m}} = 1 \text{ rad/s}\]\[ v_f = r \omega_f = 2 \omega_f \implies \omega_f = \frac{v_f}{r} = \frac{8 \text{ m/s}}{2 \text{ m}} = 4 \text{ rad/s}\]

Step 3 - Use the angular kinematic equation

Using the formula for angular velocity with constant angular acceleration:\[\omega_f = \omega_i + \alpha t\]Plug in the known values and solve for \(\alpha\):\[4 \text{ rad/s}= 1 \text{ rad/s} + (3 \text{ rad/s}^2) (1 \text{ s})\]Check if this aligns with the provided constant angular acceleration, which it does, confirming our previous steps.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object navigates around a circular path. It is represented by the symbol \( \omega \) (omega). This quantity describes the rate of change in the angular position of an object with respect to time.

The formula to find the angular velocity when you know the linear velocity \(v\) and the radius \(r\) is:

\[ \omega = \frac{v}{r} \]
In the given problem, the particle moves along the circumference of a circle with a radius of \(2 \text{ m}\). Initially, we use the linear velocity (which transitions to its angular counterpart using the radius) to find that the initial angular velocity was \(1 \text{ rad/s}\). After one second, with the particle's speed increased to \(8 \text{ m/s}\), the angular velocity becomes \(4 \text{ rad/s}\).
Hence, understanding how linear and angular quantities relate helps you comprehend the speed of rotation from the peripheral velocity of a point in the rotating body.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is represented by the symbol \( \alpha \) (alpha). This concept comes into play especially when an object's rotation speed is not constant. Angular acceleration tells us how quickly the object is speeding up or slowing down in its rotational motion.

The angular acceleration formula when the angular velocity changes over time is:

\[ \alpha =\frac{\Delta \omega}{\Delta t} \]
In our problem, the constant angular acceleration is given as \(3 \text{ rad/s}^2\). This means every second, the angular velocity increases by \(3 \text{ rad/s}\). Between initial and one second later, using the angular kinematic equation:

\[ \omega_f = \omega_i + \alpha t \]
Plugging in the values, \(4 \text{ rad/s} = 1 \text{ rad/s} + (3 \text{ rad/s}^2)(1 \text{ s})\), it verifies the given angular acceleration, confirming our calculations.

Kinematics

Kinematics involves the study of motion without considering the forces that cause motion. For angular motion, similar kinematic equations apply as in linear motion, adapted for rotational quantities. These equations help us predict an object's future position, velocity, or acceleration if we know the initial conditions and the rates of change.

For our particle, we use both linear and angular kinematic equations. The transition between linear and rotational motion can be concisely derived using the radius of the circle.

Key kinematic equation used in angular motion:

\[ \omega_f = \omega_i + \alpha t \]
Here, this equation was pivotal in connecting the dots between initial and final states, helping you see how the concepts of velocity and acceleration intertwine with time in a rotational framework. Applying kinematic principles correctly allows for accurate predictions of future states of motion.