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The example question is:

"a clock contains a pendulum arm with a mass of $3.5 \,kg$. If the spring constant of the arm is $1.0\, N/m$ and the maximum amplitude of the arm is $45$ cm, calculate the energy of the system and the maximum speed of the mass."

The solution they give goes as follows:

$$\mathcal{E}=kA^2, \quad \text{(total energy = spring constant x amplitude^2)}.$$

The result is $2.0 \times 10^{-1} J$. Next they use the equation $\mathcal{E} = (mv^2)/2 + (kx^2)/2$, substitute $2.0 \times 10^{-1} J$ for $\mathcal{E}$, set $x$ to zero and solve for $v$.

What I don't understand is the first part of the solution. In the previous example the equation $\mathcal{E} = (kA^2)/2$ was used for the total energy of the system. Where did the equation $\mathcal{E} = kA^2$ come from?

Sebastiano
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bshearer
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    I am not clear as to the configuration. Is this a "simple" pendulum with the string replaced by a spring or a mass oscillating vertically at the end of a spring or . . . . .? – Farcher Mar 24 '19 at 13:17
  • As for the first part: what is the velocity of the arm when the amplitude is maximum? Regarding the second part, if no other forces are relevant (e.g., gravity if the movement was vertical, or friction), the potential energy of the spring will be transformed in what? – Ertxiem - reinstate Monica Mar 24 '19 at 23:01
  • after using E = kA^2 to get 2.0x10^-1 J , the solution reads "Realizing that when the potential energy of the pendulum is zero when it has reached its maximum speed, we have:" and then proceeds to use E = (mv^2)/2 + (kx^2)/2 with x set to zero to solve for v. there's no other context given. what I'm stumped on is why the equation for total energy is E = kA^2, shouldn't it be E = (kA^2)/2 ? – bshearer Mar 25 '19 at 00:36

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