A horizontal spring with k = 45 N/m is attaches a 1.0kg block to a spring hung from the ceiling. A block is pulled down to a spot that is 30.0cm from the spring's rest position. What is it's new acceleration?

To calculate the new acceleration of the block, we can use Hooke's Law and Newton's laws of motion.

Step 1: Calculate the spring force
The spring force can be calculated using Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position. The formula for Hooke's Law is:

F = -kx

Where F is the spring force, k is the spring constant, and x is the displacement from the equilibrium position.

Given that the spring constant is k = 45 N/m and the displacement is x = 30.0 cm = 0.3 m, we can calculate the spring force:

F = -kx
F = -(45 N/m)(0.3 m)
F = -13.5 N

Step 2: Calculate the net force
The net force acting on the block is the sum of the spring force and the gravitational force. Since the block is hung from the ceiling, the gravitational force is equal to the weight of the block:

F_gravitational = m * g

Where m is the mass of the block and g is the acceleration due to gravity.

Given that the mass of the block is m = 1.0 kg and the acceleration due to gravity is g = 9.8 m/s^2, we can calculate the gravitational force:

F_gravitational = (1.0 kg) * (9.8 m/s^2)
F_gravitational = 9.8 N

The net force is the sum of the spring force and the gravitational force:

F_net = F_spring + F_gravitational
F_net = -13.5 N + 9.8 N
F_net = -3.7 N

Step 3: Calculate the new acceleration
According to Newton's second law of motion, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

F_net = m * a

We can rearrange this equation to solve for the acceleration:

a = F_net / m

Plugging in the values:

a = (-3.7 N) / (1.0 kg)
a = -3.7 m/s^2

Therefore, the new acceleration of the block is -3.7 m/s^2.