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Question: A body of mass \(400g\) executes SHM of amplitude \(20cm\). If the time period is \(0.2\sec \) then ...

A body of mass 400g400g executes SHM of amplitude 20cm20cm. If the time period is 0.2sec⁡0.2\sec then find the total energy of the system?

Explanation

Solution

Hint We have a simple body executing the simple harmonic motion and by applying the formula of energy which is12mω2A2\dfrac{1}{2}m{\omega ^2}{A^2}, we will be able to find the total required energy for the system. And while solving this we have to keep in mind to change the units of the needed entities.
Formula used
The total energy of the system,
E=12mω2A2E = \dfrac{1}{2}m{\omega ^2}{A^2}
Here,
EE, will be the energy
mm, will be the mass
ω\omega , will be the angular velocity
AA, will be the area
Also, the formula for the angular velocity will be,
ω=2πT\omega = \dfrac{{2\pi }}{T}
Here,
TT, will be the time period.

Complete Step By Step Solution So we have to find the total energy, for this, we will use the formula for the total energy mentioned in the formula.
E=12mω2A2E = \dfrac{1}{2}m{\omega ^2}{A^2}
So the above equation can also be written as,
E=12m(2πT)2A2E = \dfrac{1}{2}m{\left( {\dfrac{{2\pi }}{T}} \right)^2}{A^2}
Now on substituting the values, we get
Also while putting the values we had changed the units,
The equation will be like,
⇒2π2(400×10−3kg)(20×10−2m)20.20s\Rightarrow \dfrac{{2{\pi ^2}\left( {400 \times {{10}^{ - 3}}kg} \right){{\left( {20 \times {{10}^{ - 2}}m} \right)}^2}}}{{0.20s}}
Now on simplifying the above equations, we get
⇒1.577J\Rightarrow 1.577J

Therefore, 1.577J1.577J energy is required for the system.

Note Mechanical energy, the aggregate of the active energy, or energy of movement, and the possible energy, or energy put away in a framework because of the situation of its parts. Mechanical energy is consistent in a framework that has just gravitational powers or in a generally admired framework—that is, one lacking dissipative powers, for example, erosion and air obstruction, or one in which such powers can be sensibly ignored. In this manner, a swinging pendulum has its most prominent dynamic energy and least possible energy in the vertical situation, in which its speed is most noteworthy and its stature least; it has its most un-motor energy and most prominent likely energy at the limits of its swing, in which its speed is zero and its tallness is most noteworthy.