A particle having m mass with a simple hermonic motion and have instantaneous displacement x(t)=4cosπt +3sinπt . so what will be the maximum kinetic energy of the particle?

Question

Answer

1/2m|A^2w^2|

Answer

125

Answer

Understanding the Motion of the Particle
The displacement of the particle is given by the equation:
x(t) = 4cos(πt) + 3sin(πt).
This represents a simple harmonic motion (SHM), where the particle oscillates about the equilibrium position.
Calculating Velocity
To find the maximum kinetic energy (KE), we first need to determine the velocity of the particle:
- The velocity $ v(t) $ is the derivative of the displacement $ x(t) $:
$ v(t) = \frac{dx}{dt} = -4πsin(πt) + 3πcos(πt) $.
Maximum Velocity
Next, we need to find the maximum value of $ v(t) $:
- The maximum velocity occurs when the absolute value of $ v(t) $ is at its peak.
- The amplitude $ A $ of the motion can be found using the formula:
$ A = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = 5 $.
- The maximum velocity $ V_{\text{max}} $ in SHM is given by $ V_{\text{max}} = Aω $ where $ ω = 2πf $ and $ f = 1 $ (frequency).
Thus, $ V_{\text{max}} = 5 \cdot π $.
Maximum Kinetic Energy
The kinetic energy of the particle is calculated using the formula:
- $ KE = \frac{1}{2}mv^2 $.
- Substituting $ v = V_{\text{max}} $:
$ KE_{\text{max}} = \frac{1}{2}m(5π)^2 = \frac{1}{2}m(25π^2) = \frac{25π^2}{2}m $.
Conclusion
Therefore, the maximum kinetic energy of the particle is:
$ KE_{\text{max}} = \frac{25π^2}{2}m $. This value indicates the energy of the particle at its maximum velocity during the harmonic motion.

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