Final answer:
The force constant of the spring is 422.22 N/m. The magnitude force needed to stretch the spring 9.00 cm from its unstretched length is approximately 38.0 N. The work done to compress the spring 4.00 cm from its unstretched length is 0.3388 J and the force needed to compress the spring this distance is approximately 16.9 N.
Step-by-step explanation:
To find the force constant of the spring, we can use the formula for work done on a spring, which is given by:
W = (1/2)kx^2
Where W is the work done on the spring, k is the force constant, and x is the displacement of the spring from its unstressed length. Rearranging the formula, we have:
k = 2W/x^2
Substituting the given values, we have:
k = 2(19.0 J)/(0.09 m)^2 = 422.22 N/m
So, the force constant of the spring is 422.22 N/m.
To calculate the magnitude force needed to stretch the spring 9.00 cm from its unstretched length, we can use Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from equilibrium. The formula for Hooke's Law is given by:
F = -kx
Where F is the force exerted by the spring, k is the force constant, and x is the displacement of the spring. Rearranging the formula, we have:
F = -kx = -(422.22 N/m)(0.09 m) = -37.9998 N ≈ -38.0 N
Therefore, the magnitude force needed to stretch the spring 9.00 cm from its unstretched length is approximately 38.0 N.
To calculate the work done to compress the spring 4.00 cm from its unstretched length, we can use the same formula for work done on a spring as before. Substituting the given values, we have:
W = (1/2)kx^2 = (1/2)(422.22 N/m)(-0.04 m)^2 = 0.3388 J
So, the work done to compress the spring 4.00 cm from its unstretched length is 0.3388 J.
To calculate the force needed to compress the spring this distance, we can use Hooke's Law again. Rearranging the formula, we have:
F = -kx = -(422.22 N/m)(-0.04 m) = 16.888 N ≈ 16.9 N
Therefore, the force needed to compress the spring 4.00 cm from its unstretched length is approximately 16.9 N.