Question

The equation for displacement of a particle at time \(\mathrm{t}\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). If the mass of the particle is \(5 \mathrm{gm}\), then the total energy of the particle is \(\ldots \ldots \ldots\) erg (A) 250 (B) 125 (C) 500 (D) 375

Short answer

Total energy of the particle is given by the expression: \[Total\:Energy = \frac{1}{2}(0.005)((-6\sin(2t) + 8\cos(2t))^2) + (0.005)(9.81)(3\cos(2t) + 4\sin(2t))\]. None of the options (A, B, C, D) are correct, as the total energy is not a constant value but a function of time t.

Step-by-step solution

Find the velocity of the particle

To find the velocity, we need to differentiate the displacement equation with respect to time: \[\frac{dy}{dt} = \frac{d}{dt}(3\cos(2t) + 4\sin(2t))\] Apply the chain rule and differentiate each term: \[\frac{dy}{dt} = -6\sin(2t) + 8\cos(2t)\] This is the velocity of the particle.

Calculate the kinetic energy

Now, we need to calculate the kinetic energy using the formula: \[KE = \frac{1}{2}mv^2\] First, we have the mass in grams, so we need to convert it to kg: \[m = 5gm = 0.005kg\] Now, square the velocity: \[\left(\frac{dy}{dt}\right)^{2} = (-6\sin(2t) + 8\cos(2t))^2\] Substitute the mass and velocity into the kinetic energy formula: \[KE = \frac{1}{2}(0.005)((-6\sin(2t) + 8\cos(2t))^2)\]

Calculate the potential energy

To calculate the potential energy, first, find the height, which in this case is the displacement y: \[h = y = 3\cos(2t) + 4\sin(2t)\] Now, substitute the mass and height into the potential energy formula: \[PE = (0.005)(9.81)(3\cos(2t) + 4\sin(2t))\]

Calculate the total energy

Finally, add the kinetic and potential energies to find the total energy of the particle: \[Total\:Energy = KE + PE\] \[Total\:Energy = \frac{1}{2}(0.005)((-6\sin(2t) + 8\cos(2t))^2) + (0.005)(9.81)(3\cos(2t) + 4\sin(2t))\] The total energy is not a constant value as it is a function of t. So, none of the answer options (A, B, C, D) are correct since they all represent constant values.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. In mathematical terms, it is expressed using the formula: \[ KE = \frac{1}{2} mv^2 \] Where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

To calculate kinetic energy, first ensure the mass is in the correct unit, typically kilograms, even if it is initially given in grams. This conversion is essential to maintain consistency in units, especially for systems using SI units.
Next, find the velocity of the particle by differentiating its displacement function relative to time. In this context, the displacement function is \( y = 3\cos(2t) + 4\sin(2t) \). After differentiation, the velocity function is \( -6\sin(2t) + 8\cos(2t) \). Squaring this velocity and substituting the mass into the kinetic energy formula provides the kinetic energy as a function of time \( t \). This formula tells us how kinetic energy varies as the particle moves around.

Potential Energy

Potential energy is the stored energy in an object because of its position or configuration. In the framework of harmonic motion, potential energy is typically calculated using the expression: \[ PE = mgh \] Where:

• \( m \) is the mass of the object
• \( g \) is the acceleration due to gravity (approximately \(9.81\ m/s^2\) on Earth's surface)
• \( h \) is the displacement, representing the height

For the problem, the displacement \( h \) is given by the equation \( h = y = 3\cos(2t) + 4\sin(2t) \). By substituting the mass and this expression for height into the potential energy formula, we derive the potential energy as another time-dependent expression.
We can see that both kinetic and potential energies vary with time, highlighting how different forms of energy exchange and balance in harmonic motion.

Differentiation

Differentiation is a mathematical technique used to calculate the rate at which a function changes at any point. It's a critical tool when analyzing motion since it allows us to derive quantities like velocity and acceleration from displacement functions.
For example, given the displacement function \( y = 3\cos(2t) + 4\sin(2t) \), differentiation by \( t \) helps us find the velocity, \( \frac{dy}{dt} = -6\sin(2t) + 8\cos(2t) \). This process, called differentiation, involves using rules like the chain rule to compute derivatives correctly.

• Chain Rule: For composite functions, differentiate the outer function and multiply it by the derivative of the inner function.

Understanding differentiation provides meaningful insights into how various variables change, particularly in harmonic motion where knowing speed and acceleration at any given moment is crucial for full comprehension of the system's behavior.