To determine whether you can use the reduced mass system formula for the time period between two blocks attached to a spring, let's first clarify the conditions under which this formula is applicable. The concept of reduced mass is particularly useful in systems where two bodies interact through a central force, such as gravitational or spring forces. In your case, with a spring fixed at one end and two masses (m and M) attached, we need to analyze the system carefully.

Understanding the System Configuration

In the problem you mentioned from H.C. Verma, you have a spring fixed at one end with a mass (m) attached to the free end, and another mass (M) attached to mass (m). This creates a two-body system where both masses can move, but the spring's fixed end means that the spring force acts on mass (m) while mass (M) is dependent on the motion of mass (m).

Conditions for Using Reduced Mass

The reduced mass formula is typically used when:

• The two masses are interacting through a central force.
• The system can be simplified to a single-body problem using the reduced mass.
• The motion of the two masses can be described in terms of their relative motion.

Applying the Concept to Your Problem

In your scenario, the spring force acts on mass (m), and mass (M) is influenced by the motion of mass (m). The key here is that the system can be treated as a single oscillating system if we consider the relative motion between the two masses. The effective mass for the oscillation can be calculated using the reduced mass formula:

Calculating Reduced Mass

The reduced mass (μ) is given by:

μ = (m * M) / (m + M)

This allows you to treat the two masses as a single mass oscillating about the equilibrium position of the spring. The time period (T) of the oscillation can then be calculated using the formula:

T = 2π√(μ/k)

where k is the spring constant.

Conclusion on Applicability

Yes, you can use the reduced mass system formula in this case. The spring's fixed end allows you to consider the motion of the two masses as a coupled oscillation, and by using the reduced mass, you can simplify the analysis to find the time period of the oscillation. This approach not only simplifies the calculations but also provides a clear understanding of the dynamics involved in the system.