A tension spring opposes being stretched, but offers no resistance to being compressed

(think of a slinky!). A tension spring, with an equilibrium length of 1.25 m, is suspended
by one end from a ceiling at a height of 2.5 m above the ground; a mass of 100 g is
attached to the other end of the spring.
(a) If the mass is initially held and released from a height of 1.5 m above the ground
describe how the gravitational potential energy, kinetic energy and elastic potential
energy of the system change as the mass fall towards the ground. (3 points)
(b) What is the smallest spring constant that will prevent the mass from touching the
floor? (2 points)

(a) When the mass is initially held and released from a height of 1.5 m above the ground, it starts to fall due to gravity. As it falls, its gravitational potential energy decreases because it is getting closer to the ground. At the same time, its kinetic energy increases because it is gaining speed as it falls.

As the mass falls, the tension in the spring increases and it starts to stretch. The spring gains elastic potential energy as it is being stretched. This energy comes from the gravitational potential energy of the falling mass.

At some point, when the mass is about to reach the equilibrium position of the spring (1.25 m), the gravitational potential energy is at its lowest, almost equal to zero. At this point, the kinetic energy of the mass is at its highest because it has reached its maximum speed. The elastic potential energy of the spring is also at its highest because it is fully stretched.

As the mass continues to fall below the equilibrium position of the spring, the gravitational potential energy starts to increase again because it is getting farther from the ground. The kinetic energy decreases as the speed of the mass decreases. The elastic potential energy of the spring also decreases as it loses its stretch.

When the mass finally reaches the ground, its gravitational potential energy is at its highest because it is at its maximum height. The kinetic energy is equal to zero because the mass has come to a stop. The elastic potential energy of the spring is also equal to zero because it has returned to its equilibrium length.

(b) The smallest spring constant that will prevent the mass from touching the floor can be calculated using the formula for gravitational potential energy:

Ep = m * g * h

Where:
Ep is the gravitational potential energy
m is the mass (0.1 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height (1.5 m)

Setting the gravitational potential energy equal to the elastic potential energy of the spring at its maximum stretch, we can calculate the spring constant using the formula for elastic potential energy:

Ee = (1/2) * k * x^2

Where:
Ee is the elastic potential energy at maximum stretch (equal to m * g * h)
k is the spring constant (unknown)
x is the maximum stretch of the spring (equal to the equilibrium length of the spring, 1.25 m)

Setting Ee equal to Ep and solving for k:

(m * g * h) = (1/2) * k * x^2

(0.1 kg) * (9.8 m/s^2) * (1.5 m) = (1/2) * k * (1.25 m)^2

We can solve this equation to find the value of k, which will be the smallest spring constant that will prevent the mass from touching the floor.