28. Block on Incline In Fig. 16-35, a block weighing 14.0 N, which slides without friction on a

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28. Block on Incline In Fig. 16-35, a block weighing 14.0...

28. Block on Incline In Fig. $16-35$, a block weighing $14.0 \mathrm{~N}$, which slides without friction on a $40.0^{\circ}$ incline, is connected to the top of the incline by a massless spring of unstretched length $0.450 \mathrm{~m}$ and spring constant $120 \mathrm{~N} / \mathrm{m}$. (a) How far from the top of the incline does the block stop? (b) If the block is pulled slightly down the incline and released, what is the period of the resulting oscillations?

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from its unstressed length, expressed mathematically as F = -kx. This relationship is crucial when analyzing systems involving springs because it provides a linear model for the restoring force, which in turn is used to find equilibrium positions and to analyze oscillatory motion.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes periodic oscillations where the restoring force on an object is proportional to its displacement and directed towards the equilibrium position. In the context of a block on an incline connected to a spring, once displaced from equilibrium, the system will oscillate with a period that depends on the mass of the block and the spring constant, analogous to T = 2??(m/k).

Gravitational Force Decomposition

On an inclined plane, the weight of an object is split into two components: one perpendicular to the surface and one parallel to it. The parallel component, calculated as mg sin?, is responsible for driving the motion along the incline, while the perpendicular component, mg cos?, influences the normal force. This decomposition is essential for analyzing forces acting on objects on slopes.

Static Equilibrium

Static equilibrium occurs when the net force on an object is zero. For a block on an incline attached to a spring, equilibrium is achieved when the component of gravitational force along the plane exactly balances the spring's restoring force. Understanding equilibrium conditions allows one to determine the displacement where the system settles without acceleration.

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