Animated Solution for Physics - Laws of Motion: System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass $2m$ and $m$ just after the string is cut will be (2006)

The physics of non-inertial frames is like stepping into a parallel universe where invisible hands push and pull objects. This problem is a masterpiece of classical mechanics because it combines relative motion, pseudo forces, and a brilliant "dual friction" trap that catches many students off guard. Let's embark on this thrilling journey to find the relative acceleration of the block.

Imagine you are standing on the circular disc. Suddenly, the entire disc accelerates violently to the left at $a=25{\text{ m/s}}^2$. From your perspective on the disc, the block doesn't just sit there; it feels a massive invisible push to the right. This is the pseudo force.

Because we are analyzing the motion from the non-inertial frame of the accelerating disc, Newton's laws demand that we apply this fictitious force. The magnitude is simply the mass of the block times the acceleration of the frame ($ma$), and its direction is exactly opposite to the frame's acceleration.

Resolving the Forces

The block wants to fly straight to the right, but it can't. It is physically trapped inside a groove that is angled at $\theta$ to the horizontal. To understand what happens next, we must resolve our pseudo force into two critical components:

1. The Driving Component: The component of the pseudo force acting along the groove is $ma\cos \theta$. This is the engine. It's the force desperately trying to slide the block forward through the channel. 2. The Crushing Component: The component acting perpendicular to the groove is $ma\sin \theta$. This force doesn't cause motion; instead, it violently smashes the block against the side wall of the groove.

Here is where the problem separates the masters from the novices. When we think of friction, we usually just think of an object sliding on a floor. But look closely at the block's physical reality! It is in contact with two distinct surfaces.

First, it rests on the horizontal floor of the disc. Gravity pulls it down, so the floor pushes up with a vertical normal force $N_1=mg$. This generates our first frictional force:

Second, remember that crushing component of the pseudo force? It presses the block against the vertical side wall of the groove. The wall pushes back with a horizontal normal force $N_2=ma\sin \theta$. This generates our second frictional force:

Both of these frictional forces act in the same direction—backwards along the groove—desperately trying to oppose the block's forward motion.

Now, we bring it all together using Newton's Second Law along the axis of the groove. The net force dictates the relative acceleration $a_r$ of the block.

The driving force is fighting against the two frictional forces:

Substituting our expressions for friction, we reveal the raw structure of the physics:

Notice something beautiful? Every single term contains the mass $m$. The mass of the block is completely irrelevant to its kinematic fate! We can divide the entire equation by $m$ to get our master kinematic equation:

Before we plug in the numbers, we need to unlock the geometry. We are given $\cos \theta =\frac{4}{5}$. Using the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$, or simply recognizing the classic 3-4-5 right triangle, we immediately know that $\sin \theta =\frac{3}{5}$.

Now, we unleash our arsenal of values: $a=25{\text{ m/s}}^2$, $g=10{\text{ m/s}}^2$, and $\mu =\frac{2}{5}$.

Let's execute the atomic computations: - The driving term: $25\times \frac{4}{5}=20$ - The floor friction term: $\frac{2}{5}\times 10=4$ - The wall friction term: $\frac{2}{5}\times 25\times \frac{3}{5}=6$

Bringing it all home:

The block accelerates along the groove at exactly $10{\text{ m/s}}^2$ relative to the disc. By carefully mapping the non-inertial forces and respecting the physical constraints of the dual surfaces, a complex 3D mechanics problem collapses into an elegant and satisfying solution.