A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 c/m when 𝓍 = -5.0 cm What is the block's position when the acceleration is maximum?

Verified step by step guidance

1. Convert all given quantities to SI units for consistency. The mass of the block is 100 g = 0.1 kg, the spring constant is 2.5 N/m, the velocity is 20 cm/s = 0.2 m/s, and the position is x = -5.0 cm = -0.05 m.
2. Recall that the acceleration of a block in simple harmonic motion is given by the formula: $a=\; -\omega x^2$, where $\omega$ is the angular frequency. The acceleration is maximum when the block is at its maximum displacement (amplitude).
3. Determine the angular frequency using the formula: $\omega =\; \surd (k/m)$, where $k$ is the spring constant and $m$ is the mass of the block.
4. The maximum displacement (amplitude) can be found using the conservation of energy principle: the total mechanical energy is constant and is the sum of kinetic energy and potential energy. Use the formula: $E=\; 1/2k{A}^2=\; 1/2k{x}^2+\; 1/2m{v}^2$, where $A$ is the amplitude, $x$ is the position, and $v$ is the velocity.
5. Once the amplitude is determined, the block's position when the acceleration is maximum is equal to the amplitude, as acceleration is maximum at the maximum displacement in simple harmonic motion.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium and acts in the opposite direction. This motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and phase.

Acceleration in SHM

In Simple Harmonic Motion, the acceleration of the oscillating object is maximum when it is at its maximum displacement from the equilibrium position. This is because the restoring force, which causes the acceleration, is greatest at these points. The relationship between acceleration, displacement, and the spring constant can be expressed as a = -k/m * x, where k is the spring constant, m is the mass, and x is the displacement.

Equilibrium Position

The equilibrium position in a spring-mass system is the point where the net force acting on the mass is zero. For a spring, this is the position where the spring is neither compressed nor stretched. In the context of oscillation, the maximum displacement from this point corresponds to the maximum potential energy and minimum kinetic energy, while the maximum acceleration occurs at these extreme positions.