To solve this problem, we start by analyzing the relationship between total energy and amplitude for oscillations. The total mechanical energy $E$ in simple harmonic motion is proportional to the square of the amplitude $A$. This can be described by the equation:

$E \propto A^2$

Or, more specifically,

$E = kA^2$

where $k$ is a constant of proportionality that depends on the mass and spring constant or other system parameters.

Step 1: Use the given information

We know that:

• The total energy $E = 40 \, \text{J}$ when the amplitude $A = 0.50 \, \text{m}$.

Substitute these values into the equation to find the constant $k$:

$40 = k(0.50)^2$ $40 = k(0.25)$ $k = \frac{40}{0.25} = 160$

Thus, the relationship between energy and amplitude is:

$E = 160A^2$

Step 2: Plot the equation

The equation $E = 160A^2$ is a quadratic equation, where the total energy increases with the square of the amplitude.

Energy values at various amplitudes:

• For $A = 0 \, \text{m}$, $E = 160(0)^2 = 0 \, \text{J}$
• For $A = 0.25 \, \text{m}$, $E = 160(0.25)^2 = 160(0.0625) = 10 \, \text{J}$
• For $A = 0.50 \, \text{m}$, $E = 40 \, \text{J}$ (as given)
• For $A = 0.75 \, \text{m}$, $E = 160(0.75)^2 = 160(0.5625) = 90 \, \text{J}$
• For $A = 1.00 \, \text{m}$, $E = 160(1.00)^2 = 160 \, \text{J}$

Step 3: Graph the relationship

Now, you can sketch the graph of $E = 160A^2$. The amplitude $A$ is on the x-axis (from 0 to 1.00 m), and the total energy $E$ is on the y-axis. The y-axis should be scaled to go from 0 to at least 160 J. The graph will be a parabola opening upwards, starting at $E = 0$ when $A = 0$, passing through $E = 40 \, \text{J}$ at $A = 0.50 \, \text{m}$, and reaching $E = 160 \, \text{J}$ at $A = 1.00 \, \text{m}$.

Would you like me to generate this graph for you?

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5 Related Questions:

1. How does the energy change if the amplitude of oscillation is doubled?
2. What happens to the total energy if the mass of the child is doubled, keeping amplitude constant?
3. How is the constant $k$ affected by changes in the spring constant or length of a pendulum?
4. How would the energy vs. amplitude graph differ for a damped oscillator?
5. What is the relationship between frequency and amplitude in simple harmonic motion?

Tip: The total energy in oscillatory systems is always proportional to the square of the amplitude for simple harmonic motion.