91Ó°ÊÓ

Hooke's Law is a fundamental principle in physics that describes how the force exerted by a spring is related to its compression or extension. Essentially, Hooke's Law can be expressed by the formula:

• \( F_s = kx \)

Where \( F_s \) is the spring force, \( k \) is the spring constant, and \( x \) is the compression or extension of the spring. This law shows that the force exerted by a spring is directly proportional to the distance it is compressed or extended from its natural length.

In the context of our example, when a force of \( 3mg \) compresses the spring to \( \frac{4mg}{k} \), Hooke's Law helps us determine the spring force. We can see that the spring exerts an upward force of \( 4mg \) when compressed, as the force calculated from Hooke’s Law (\( F_s = 4mg \)) balances the additional force applied. This ensures stability until the configuration changes, like when the mass is released.

The concept of equilibrium of forces is pivotal in understanding how forces interact in a system. At equilibrium, the total forces acting on an object are balanced. This means that the net force is zero, resulting in a state of rest or constant velocity for the object.

In our problem, we initially had a situation where the upper block with mass \( m \) was held in place by a downward force of \( 3mg \) in addition to the gravitational force \( mg \). The spring, compressed as a result, exerts an upward force equal to \( 4mg \). Here, the equilibrium condition can be visualized as:

• Total downward force = Total upward force
• \( 3mg + mg = 4mg \)
• Thus, balance is achieved before release

Understanding this balance ensures that when the block is released, the spring force can act without interference, aiming to lift the system or adjust the equilibrium state.

Gravitational force is the attractive force that pulls objects towards the center of a massive body, like Earth. It's the force that gives weight to physical objects and is calculated using the formula:

• \( F_g = mg \)

Here, \( F_g \) represents the gravitational force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth).

In our exercise, gravitational force acts on both blocks making up the system. For the lower block to lift off the table when the spring is released, the spring force needs to be strong enough to counteract and exceed the combined gravitational force affecting both blocks. This requirement is modeled in:

• \( 4mg \geq (M+m)g \)

Gravitational force, thus, dictates the minimum spring force necessary for the lower block to be lifted.

In many physics problems, particularly those dealing with force and motion, determining the ratio of different quantities can provide deeper insight.

For our specific scenario, we are interested in finding out the greatest possible value for the ratio \( \frac{M}{m} \). Starting with the inequality derived from overbalancing forces:

• \( 4mg \geq (M+m)g \)

This simplifies to:

\( 4m - m \geq M \)

From there, we solve for \( M \) as:

• \( 3m \geq M \)

By dividing both sides by \( m \), we obtain:

• \( \frac{M}{m} \leq 3 \)

Which tells us the maximum possible ratio of the masses is 3. This simple calculation plays a crucial role in determining systems’ responses to different physical interactions.