91Ó°ÊÓ

When thinking about work done by forces, it's important to know that work is defined as the energy transferred to or from an object via a force acting upon it. The basic formula for work is:
\[ W = F \times d \times \text{cos}(\theta) \]
where \(W\) is the work done, \(F\) is the force, \( d\) is the displacement, and \(\theta\) is the angle between the force and displacement direction.
In the context of the exercise, we calculated the work done by the gravitational force while the block compresses the spring. This work can be expressed as: \( W_g = mgh \), where \(m\) is the mass, \(g\) is the gravitational acceleration, and \(h\) is the height or displacement (0.12 m in this case).
For the spring force, Hooke's Law tells us that the force exerted by the spring is proportional to the distance it is compressed: \( F_s = -kx \). The work done by the spring force during compression is given by: \( W_s = \frac{1}{2}kx^2 \), with \(k\) being the spring constant and \(x\) the compression length.

The spring constant \( k \) is a measure of a spring's stiffness. A higher spring constant indicates a stiffer spring, which requires more force to compress or extend it. The formula relating the force \( F \) exerted by a spring to its displacement from the equilibrium position \( x \) is:
\[ F = kx \]
In the exercise, the spring constant \( k \) was given as 250 N/m. This constant is crucial for calculating both the work done by the spring and the potential energy stored in it. When the block compresses the spring by 0.12 meters, the spring exerts a restoring force, and the work done by this force can be found using the formula \( W_s = \frac{1}{2}kx^2 \).

The conservation of energy principle states that energy cannot be created or destroyed, only converted from one form to another. In our problem, the total mechanical energy (the sum of kinetic and potential energy) of the block-spring system remains constant if we neglect friction.
Initially, the block has potential energy due to its height and kinetic energy as it falls. When the block compresses the spring, its kinetic energy is converted into elastic potential energy stored in the spring.
We use the equation:
\[ mgh = \frac{1}{2}mv^2 \]
to find the speed of the block just before it hits the spring.
When the speed at impact is doubled, the kinetic energy increases by a factor of four. Solving the equation:
\[ \frac{1}{2}mv'^2 = \frac{1}{2}ky^2 \]
, we relate the increased kinetic energy to the new compression of the spring.

Kinetic energy is the energy of an object due to its motion. The formula for kinetic energy (KE) is:
\[ KE = \frac{1}{2}mv^2 \]
where \(m\) is the mass and \(v\) is the velocity of the object.
In the exercise, finding the speed just before the block hits the spring involves converting gravitational potential energy into kinetic energy. By applying the equation:
\[ mgh = \frac{1}{2}mv^2 \]
and solving for \(v\), we determined how fast the block was moving just before it made contact with the spring.

Potential energy (PE) is the stored energy of an object due to its position in a force field, like gravity or a spring. In our scenario, we mainly deal with gravitational potential energy and elastic potential energy. Gravitational potential energy is calculated as:
\[ PE = mgh \]
where \(m\) is the mass, \(g\) is the gravitational acceleration, and \(h\) is the height. Elastic potential energy stored in a compressed or stretched spring is given by Hooke's Law:
\[ PE = \frac{1}{2}kx^2 \]
During the block’s fall, its gravitational potential energy converts to kinetic energy, which then converts to elastic potential energy as the spring compresses. Understanding these conversions helps in solving for variables like speed before impact and maximum compression when speed is doubled.