Energy in Simple Harmonic Motion

[chaotixmonjuish]

A 1.3 kg object oscillates with simple harmonic motion on a spring of force constant 410.0 N/m. The maximum speed is 0.7 m/s. What is the total energy of the object and the spring?

Do I just set it up like this:

1/2*410*x^2=1/2*1.3*.7^2 to get the max x
Then do I just add them up

 

[Shooting Star]

No. Both are equal to the total energy. Use just one of them.

Easier would be to write E_tot = (1/2)m(v_max)^2, since m and v_max is given.

 

[ogb p]

?

Yes, that is the correct approach. To find the total energy, we need to consider the potential energy stored in the spring and the kinetic energy of the oscillating object. In simple harmonic motion, the potential energy is given by 1/2*k*x^2 where k is the force constant and x is the displacement from equilibrium. In this case, the maximum displacement can be found using the maximum speed given as v_max = ω*A where ω is the angular frequency and A is the amplitude. So, we can write the equation as 1/2*k*A^2=1/2*m*v_max^2 where m is the mass of the object.

Plugging in the values given, we get 1/2*410*A^2=1/2*1.3*0.7^2. Solving for A, we get A=0.067 m. Now, to find the total energy, we simply add the potential energy and kinetic energy together. So, the total energy of the object and the spring is 1/2*410*0.067^2+1/2*1.3*0.7^2 = 2.03 J. This energy is constantly being exchanged between potential and kinetic energy as the object oscillates back and forth, demonstrating the conservation of energy in simple harmonic motion.