Animated Solution for Physics - Oscillations: A particle performs simple harmonic motion with amplitude $A$. Its speed is trebled at the instant that it is at a distance $\frac{2}{3}A$ from equilibrium position. The new amplitude of the motion is

Imagine a particle oscillating back and forth in Simple Harmonic Motion (SHM). It has a natural rhythm and a maximum displacement, which we call the amplitude, $A$.

To understand what happens when we suddenly change the particle's speed, we need to rely on the master equation that connects velocity ($v$), angular frequency ($\omega$), amplitude ($A$), and position ($x$):

Squaring both sides gives us a form that is much easier to work with algebraically:

The problem tells us to freeze time when the particle is at a specific distance from the equilibrium position: $x=\frac{2}{3}A$. At this exact moment, let's say its velocity is $v$.

We can substitute this position into our master equation to capture the initial state of the system:

Now, the twist! At this very instant, the particle receives a sudden boost, and its speed is trebled. The new speed is $3v$.

Crucially, because this happens instantaneously, the position of the particle hasn't changed. It is still at $x=\frac{2}{3}A$. However, with this massive injection of kinetic energy, the particle will now swing much further out before coming to a stop. It has a new, larger amplitude, which we will call $A^{\prime }$.

Let's write the master equation again, but this time for the new state:

We now have a system of two equations. The most elegant way to solve this and eliminate the unknowns ($v$ and $\omega$) is to divide the initial state equation by the final state equation:

Notice how beautifully $v^2$ and ${\omega }^2$ cancel out! We are left with a pure algebraic relationship:

Now, we simply cross-multiply to solve for our new amplitude, $A^{\prime }$:

Moving the term to the right side:

Taking the square root of both sides reveals our final answer:

The new amplitude of the motion is $\frac{7}{3}A$. By trebling the speed, we increased the kinetic energy by a factor of 9, which drastically expanded the boundaries of the oscillation!