91Ó°ÊÓ

When dealing with springs, understanding the spring constant, denoted by the symbol \( k \), is crucial. The spring constant is a measure of the stiffness of a spring. It's expressed in units of newtons per meter (N/m), indicating how much force is needed to stretch or compress the spring by one meter.
A high spring constant means the spring is stiff, requiring more force for the same amount of stretch compared to a spring with a lower constant. In our problem, the spring constant is given as \( 400 \, \text{N/m} \). This means that for every meter the spring is stretched or compressed, a force of 400 newtons is exerted by the spring.

• The spring follows Hooke's Law, which is reflected in the equation: \( F = kx \), where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.
• This relation helps us calculate the potential energy stored in the spring when it is displaced, expressed by the formula: \( PE = \frac{1}{2} k x^2 \).

Thus, the stiffness of the spring plays a significant role in how energy is stored and transformed in such systems.

Kinetic energy is the energy an object possesses due to its motion. In our scenario, the fake cookie sliding on a horizontal surface has a kinetic energy of \( 20.0 \, \text{J} \) at the equilibrium position.
Kinetic energy is calculated using the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.
As the cookie moves and encounters the spring, its kinetic energy is partly converted into potential energy stored in the spring, while some energy is lost due to work done against friction.

• Initial kinetic energy at equilibrium plays a crucial role in determining how far the cookie will slide. This energy is used to overcome obstacles like friction and to compress or stretch the spring.
• When the cookie comes to a stop, its kinetic energy becomes zero, indicating that all energy has been converted to spring potential or used in overcoming friction.

It's important to track how energy transforms between kinetic and other forms, like potential, as this helps us solve for variables such as distance and energy loss.

Frictional force is an essential factor in any motion-based problem, as it represents resistance between two surfaces. Here, a frictional force of magnitude \(10.0 \, \text{N} \) acts on the sliding cookie.
This force affects how far the cookie can move after being released and contributes to the work done as the cookie moves along the surface.

• Frictional force can be calculated using the equation \( F_{friction} = \mu N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force.
• In our exercise, the calculation of work done by friction (\( W_{friction} = f_d \cdot d \)) is pivotal. This work, as a form of energy loss, affects how far the cookie will travel before coming to rest.

The presence of friction is why the cookie will not return to its original kinetic energy level during its return motion, as some energy is continually lost to frictional work.