To find the work done by the spring in bringing the block to rest, we can use the formula for the potential energy stored in a spring.
The potential energy stored in a spring that is compressed or stretched by a distance x from its equilibrium position is given by:
U = (1/2) k x^2
where U is the potential energy, k is the spring constant, and x is the distance from the equilibrium position.
In this case, the block compresses the spring by a maximum distance of 5.50 cm, which is equivalent to 0.055 m. The spring constant is given as 224 N/m.
Substituting these values into the formula, we can calculate the potential energy stored in the spring:
U = (1/2) * 224 N/m * (0.055 m)^2
Once we have calculated the potential energy stored in the spring, we can find the work done by the spring in bringing the block to rest. The work done by a force is equal to the change in potential energy associated with that force. Therefore, the work done by the spring is the negative of the change in potential energy:
Work done by the spring = -ΔU
Next, let's determine the amount of mechanical energy being dissipated by the force of friction while the block is being brought to rest.
Mechanical energy refers to the sum of kinetic energy and potential energy of the object. The change in mechanical energy can be calculated by subtracting the final mechanical energy from the initial mechanical energy:
Change in mechanical energy = Final mechanical energy - Initial mechanical energy
Since the block comes to rest, it means the final kinetic energy is zero. Therefore, the change in mechanical energy is equal to the initial kinetic energy plus the initial potential energy:
Change in mechanical energy = Initial kinetic energy + Initial potential energy
To find the initial kinetic energy, we need to calculate the initial speed of the block when it hits the spring. We can use the conservation of mechanical energy principle which states that the total mechanical energy of an object is conserved when only conservative forces are acting on it.
Since the only conservative force acting on the block is the spring force, the total mechanical energy can be calculated using the formula:
Total mechanical energy = Initial kinetic energy + Initial potential energy
Since the block starts from rest, its initial kinetic energy is zero. Therefore, the initial potential energy is equal to the initial mechanical energy.
Finally, to find the speed of the block when it hits the spring, we can use the conservation of mechanical energy principle. The total mechanical energy at the start, which is equal to the initial potential energy, is equal to the total mechanical energy when the block hits the spring, which is the sum of the final kinetic energy and the potential energy stored in the compressed spring. Using this information, we can calculate the final kinetic energy and then find the speed of the block.
I hope this explanation helps you solve the problem!