91Ó°ÊÓ

Simple harmonic motion refers to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Imagine a spring-block system where a block is attached to a spring. When the block is pulled and then released, it will move back and forth around its equilibrium position. This back-and-forth motion is what we call simple harmonic motion (SHM). In SHM, the motion can be described using sine or cosine functions, and it is characterized by an amplitude (the maximum distance from the equilibrium position), a period (the time it takes to complete one full cycle), and the angular frequency.

Static friction is the frictional force that prevents two surfaces from sliding past each other. It acts when an object is stationary and ensures that it does not start moving in response to applied forces. The maximum static friction force is given by the product of the coefficient of static friction (a constant that depends on the materials in contact) and the normal force (the force perpendicular to the surfaces in contact).
For our problem, the static friction plays a crucial role. We need to find the maximum amplitude of the spring-block system's simple harmonic motion that will not cause the smaller block to slip. If the spring causes too much acceleration, the static friction will not be able to prevent slipping.

Angular frequency is a measure of how quickly an object moves through its cycle of simple harmonic motion and is denoted by \( \omega \). The angular frequency is related to the spring constant \( k \) and the total mass of the system \( m + M \). It can be calculated using the formula:
\[ \omega = \sqrt{\frac{k}{m+M}} \]
In our example, we substitute the values: the spring constant \( k = 200 \) N/m and the total mass \( m + M = 1 + 10 = 11 \) kg. This gives:
\[ \omega = \sqrt{\frac{200}{11}} \approx 4.27 \text{ rad/s} \]. This angular frequency tells us how fast the system oscillates.

In simple harmonic motion, the maximum acceleration occurs at the maximum displacement, or amplitude, from the equilibrium position. The relationship between the maximum acceleration \( a_{max} \) and amplitude A is given by:
\[ a_{max} = \omega^2 \times A \]
Given \( a_{max} = 3.92 \text{ m/s}^2 \) (calculated from the maximum static friction force) and \( \omega = 4.27 \text{ rad/s} \), we solve for the amplitude A:
\[ A = \frac{a_{max}}{\omega^2} \]
Substitution gives:
\[ A = \frac{3.92}{(4.27)^2} \approx 0.215 \text{ m} \]. The amplitude must not exceed this value to prevent the smaller block from slipping.

A spring-block system consists of a block attached to a spring, and it often oscillates in simple harmonic motion when displaced from its equilibrium position. In our problem, we have two blocks and a horizontal spring on a frictionless surface. The spring constant (stiffness) of the spring is given as 200 N/m. The goal is to ensure the smaller block, due to the static friction between the blocks, does not slip over the larger block when the system is oscillating.
This system's behavior is pivotal for understanding many physical situations where forces and motion are involved, such as vibrations in mechanical structures, the behavior of molecules, and even the basics of many instruments in physics.