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.