Question

A particle in \(\mathrm{xy}\) plane is governed by \(\mathrm{x}=\mathrm{A} \cos \omega \mathrm{t}, \mathrm{y}=\mathrm{A}\) \((1-\sin \omega \mathrm{t}) . \mathrm{A}\) and \(\omega\) are constants. What is the speed of the particle. (A) A\omegat (B) \(\mathrm{A} \omega^{2} \mathrm{t}\) (C) \(\mathrm{A} \omega\) (D) \(\mathrm{A}^{2} \omega \sin (\omega \mathrm{t} / 2)\)

Short answer

The short answer is: The speed of the particle is (C) \(A\omega\).

Step-by-step solution

Find the velocity in the x-direction

To find the velocity in the x-direction, we need to take the time derivative of the x(t) equation: \(v_x(t) = \frac{dx}{dt}\). Using the chain rule, we have: \(v_x(t) = \frac{d}{dt} [A \cos(\omega t)] = -A\omega\sin(\omega t)\)

Find the velocity in the y-direction

Similarly for the y-direction, we need to take the time derivative of the y(t) equation: \(v_y(t) =\frac{dy}{dt}\). Using the chain rule, we have: \(v_y(t) = \frac{d}{dt} [A(1 - \sin(\omega t))] = -A\omega\cos(\omega t)\)

Find the speed of the particle

Now that we have the velocity components, we can find the speed of the particle using the formula: \(\mathrm{speed} = \sqrt{v_x(t)^2 + v_y(t)^2}\) Substitute \(v_x(t) = -A\omega\sin(\omega t)\) and \(v_y(t) = -A\omega\cos(\omega t)\): \(\mathrm{speed} = \sqrt{(-A\omega\sin(\omega t))^2 + (-A\omega\cos(\omega t))^2}\)

Simplify the speed equation

Expanding and simplifying the speed equation, we get: \(\mathrm{speed} = \sqrt{(A^2\omega^2\sin^2(\omega t) + A^2\omega^2\cos^2(\omega t))}\) Factoring out \(A^2\omega^2\), we have: \(\mathrm{speed} = \sqrt{A^2\omega^2(\sin^2(\omega t) + \cos^2(\omega t))}\) Since \(\sin^2(\omega t) + \cos^2(\omega t) = 1\), the speed expression simplifies to: \(\mathrm{speed} = A\omega\) Therefore, the correct answer is (C) \(A\omega\).

Key concepts

Velocity Components

When a particle moves in the XY plane, its position can be described using functions of time, such as \(x(t)\) and \(y(t)\). These functions indicate where the particle is located at any given time along the X and Y axes. To understand motion better, we're interested in the particle's velocity components. These components are found by taking the derivative of the position functions with respect to time.

• X-direction velocity (\(v_x(t)\)): The velocity here is the rate of change of \(x(t)\) over time. The derivative gives us the speed and direction in the x-axis.
• Y-direction velocity (\(v_y(t)\)): Similar to the x-direction, this is the rate of change of \(y(t)\) over time, providing the velocity component along the y-axis.

Having these components separated helps in analyzing and understanding the two-dimensional motion in a plane.

Time Derivative

The time derivative is a mathematical tool that helps us understand how a quantity changes over time. In the context of particle motion, it is used to find velocities from position functions. When the position of a particle is given by equations like \[x = A\cos(\omega t) \ y = A(1 - \sin(\omega t))\], we take the derivative of these with respect to time \(t\):

• For x: The derivative \(\frac{dx}{dt} = -A\omega\sin(\omega t)\). This shows that the velocity depends on both the amplitude (A) and the angular frequency (\(\omega\)).
• For y: The derivative \(\frac{dy}{dt} = -A\omega\cos(\omega t)\). This derivative also reveals a similar dependency.

The derivatives help us understand the instantaneous speed and direction of the particle in both the x and y directions.

Speed Calculation

Once the velocity components are found, calculating speed involves a bit of straightforward math. Speed is a scalar quantity representing how fast an object is moving, regardless of its direction. To find speed from velocity components (\(v_x(t)\) and \(v_y(t)\)), use the formula:\[\text{speed} = \sqrt{v_x(t)^2 + v_y(t)^2}\]Here, we substitute the derived velocity components:

• X-component: \(v_x(t) = -A\omega\sin(\omega t)\)
• Y-component: \(v_y(t) = -A\omega\cos(\omega t)\)

Putting these into the speed formula, it simplifies down to \(A\omega\), showing that speed remains constant, independent of time and direction as long as the motion parameters are constant.

Trigonometric Identities

Trigonometric identities are key to simplifying expressions involving sine and cosine functions, especially when dealing with periodic motion like oscillations. One of the most valuable identities used is:\[\sin^2(\theta) + \cos^2(\theta) = 1\]In the context of this exercise, the identity helps us simplify the expression for speed. After deriving the velocity components and substituting them into the speed formula, we encounter:\[A^2\omega^2(\sin^2(\omega t) + \cos^2(\omega t))\]Using the trigonometric identity, we find:\[\sin^2(\omega t) + \cos^2(\omega t) = 1\]This substitution simplifies the expression drastically, leading directly to \(A\omega\). Understanding these identities enables quick and easy simplification of complicated trigonometric expressions in physics and mathematics.