91Ó°ÊÓ

Static friction is the force keeping two surfaces at rest relative to each other. In simpler terms, it prevents things from slipping. When you gently push on a book lying on the table, it doesn't move immediately. This is because static friction keeps it in place.
When considering simple harmonic motion (SHM), where a child and slab move back and forth together on a spring, static friction is crucial. If static friction is strong enough, it will keep the child on the slab even when the system is in motion.
The maximum static friction force can be calculated by the formula \( F_{friction} = \mu (m+M)g \), where \( \mu \) is the coefficient of static friction, \( m \) and \( M \) are the masses, and \( g \) is the acceleration due to gravity.

• The static friction force ensures that the child does not slip off the slab.
• A higher coefficient of static friction allows for greater forces before motion begins.

The spring constant, often denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is needed to stretch or compress the spring by a certain distance.
In our problem, the spring constant is given as \( 430 \, \mathrm{N/m} \). This means that 430 Newtons of force will stretch the spring by one meter.

• The larger the spring constant, the stiffer the spring.
• In simple harmonic motion, the spring constant helps determine the system's frequency and energy.

The force exerted by the spring when it is displaced is given by \( F_{spring} = kA \), where \( A \) is the displacement from the equilibrium position. Thus, the maximum force in our example at a displacement of \( A = 0.5 \, \mathrm{m} \) is \( 215 \, \mathrm{N} \).
Understanding the spring constant helps us predict how much force will be exerted as the system oscillates.

Weight restriction refers to the maximum weight that can be safely handled by the system—in this case, the child riding on the slab. To maintain smooth, slip-free motion, we must ensure the weight doesn't exceed the maximum static friction available.
By setting the static friction force equal to the spring force, we calculate the maximum permissible weight of the child.

• If static friction cannot match the spring force, the child will slip, and the system becomes unsafe.
• The weight restriction ensures the operation remains within safety parameters.

For this exercise, by equating forces, we determined the maximum mass for the child to be approximately \( 19.84 \, \mathrm{kg} \). This means any child weighing more than this could cause slipping, preventing the system from functioning as intended.

Displacement in the context of simple harmonic motion is the distance from the equilibrium position - the point where the spring is neither stretched nor compressed. Displacement is crucial as it determines how much potential energy is stored in the system.
In our scenario, the displacement amplitude is \( 0.50 \, \mathrm{m} \). This is the maximum distance the slab (and the child) move from their resting position.

• The amplitude of displacement contributes to the calculation of maximum spring force \( F_{spring} = kA \).
• Greater displacement results in higher energy and force being exerted.

Understanding displacement in SHM helps predict the motion characteristics such as speed and acceleration at various points during the oscillation cycle. This foundational concept guides the safe setup and operation of systems like the child and slab scenario.