A Block Of Mass 308 Kg Is Attached To A Spring Which Is Re ✓ Solved

A Block Of Mass M 308 Kg Is Attached To A Spring Which Is Resting O

A block of mass m = 3.08 kg is attached to a spring which is resting on a horizontal frictionless table. The block is pushed into the spring, compressing it by 5.00 cm, and is then released from rest. The spring begins to push the block back toward the equilibrium position at x = 0 cm. The graph shows the component of the force (in N) exerted by the spring on the block versus the position of the block (in cm) relative to equilibrium. Use the graph to answer the following questions.

Sample Paper For Above instruction

This problem involves analyzing the oscillatory motion of a mass attached to a spring on a frictionless horizontal surface. The key concepts include Hooke's law, energy conservation, and analyzing force versus displacement graphs to determine elastic potential energy and oscillation characteristics.

Understanding the System

The system consists of a mass (m = 3.08 kg) connected to an ideal spring on a frictionless surface. The initial condition involves compressing the spring by 5.00 cm from its equilibrium position (x = 0), then releasing it from rest. The goal is to understand the forces involved and the motion characteristics derived from the force-displacement graph.

Force vs. Displacement Graph Analysis

The graph depicts the component of the spring force (in Newtons) as a function of the displacement (in centimeters). In the case of ideal springs obeying Hooke’s law, the force should be linearly proportional to the displacement, expressed as F = -k x, where k is the spring constant. The graph's shape provides insights into the spring's behavior and whether the spring follows Hooke’s law perfectly.

Determining the Spring Constant (k)

From the force vs. displacement graph, the spring constant k can be determined by calculating the slope of the force versus displacement line. Specifically, if the graph is linear, the slope (ΔF/Δx) yields the value of k. For example, if the force at maximum compression (x = 5.00 cm = 0.05 m) is known from the graph, k can be calculated as:

k = -F / x

where F is the magnitude of the force at maximum displacement. This provides a crucial parameter in understanding the system's oscillatory behavior.

Energy Considerations

Since the surface is frictionless, the mechanical energy remains conserved. The potential energy stored in the spring at maximum compression is converted into kinetic energy at the equilibrium point. The initial potential energy stored in the spring is:

PE_spring = (1/2) k x^2

where x = 5.00 cm = 0.05 m. At the equilibrium position, the kinetic energy of the mass is maximum, and the spring potential energy is zero (assuming no damping).

Calculating Maximum Speed

The maximum speed of the mass can be found using energy conservation principles. At maximum compression, the velocity is zero, and all energy is stored as spring potential energy. At the equilibrium point, the spring potential energy is zero, and all energy is kinetic:

(1/2) m v_max^2 = (1/2) k x^2

Solving for v_max gives:

v_max = sqrt( (k x^2) / m )

Using this relation, once k is known from the force-displacement graph, the maximum speed can be computed.

Oscillation Period

The period of oscillation T for simple harmonic motion can be derived from the spring constant and mass:

T = 2π sqrt( m / k )

With k determined, T offers insights into the time it takes for one complete oscillation.

Discussion on the Force Graph and Nonlinearities

If the force vs. displacement graph is linear, the system behaves as a simple harmonic oscillator. Deviations from linearity could imply nonlinear spring behavior, which impacts the oscillation amplitude and period. Such behaviors include anharmonicity or potential energy curves that differ from the quadratic form typical in ideal springs.

Practical Applications and Significance

This analysis is fundamental in designing mechanical systems such as suspension systems, seismographs, and various oscillatory devices. Accurate determination of the spring constant through force graphs informs engineering decisions about the system’s robustness and response characteristics.

Conclusion

The problem encapsulates core principles of classical mechanics by connecting force-displacement graphs with energy conservation and oscillatory motion. Properly analyzing these relationships allows predicting the behavior of mass-spring systems and designing devices with desired dynamic characteristics.

References

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