An object is attached to the lower end of a 100-coil spring that is hanging from the ceiling. The spring stretches by 0.210 m. The spring is then cut into two identical springs of 50 coils each. As the drawing shows, each spring is attached between the ceiling and the object. By how much does each spring stretch?
It would be .210m/4 = .0525
why don't they show the answers for these questions...
Well, that's quite a slinky situation you've got there! Let's do some springy math.
If the original 100-coil spring stretches by 0.210 m, we can say its spring constant, k, is constant. Now, when we cut the spring in half to get two identical 50-coil springs, the total number of coils and the spring constant remain the same.
So, each of the new springs will still stretch by 0.210 m when attached between the ceiling and the object. That means each spring will stretch by 0.210 m.
I guess you could say those springs are really stretching themselves to help out the object.
To solve this problem, we can make use of Hooke's Law, which states that the force needed to stretch or compress a spring is directly proportional to the displacement from its equilibrium position.
First, let's calculate the spring constant of the original 100-coil spring. The spring constant, denoted by 'k,' determines the stiffness of the spring and is a measure of how much force is required to extend or compress the spring by a certain distance.
Given:
Number of coils, N = 100
Displacement, x = 0.210 m
Next, we need to calculate the spring constant using Hooke's Law. The formula for Hooke's Law is:
F = -kx
where F is the force applied to the spring, k is the spring constant, and x is the displacement.
Since the object is hanging, the force exerted on the spring is due to gravity and can be calculated using:
F = mg
where m is the mass of the object and g is the acceleration due to gravity.
We know that the displacement x can be written as:
x = F/k
Rearranging the equation to solve for k:
k = F/x
Now, let's find the spring constant of the original 100-coil spring:
k = (mg) / x
Assuming the mass of the object is known, substitute the values into the equation to find k.
Once we have the spring constant for the original spring, we can use it to calculate the displacement of the two identical springs.
Since they are identical, each spring would require the same force to stretch it by a certain amount. Therefore, we can use the equation:
x = F / k
Rearranging the equation to solve for x:
x = (mg) / k
Substituting the known values, including the new number of coils (50) and the previously calculated spring constant k:
x = (mg) / k
Finally, calculate the value of x for each of the two identical 50-coil springs by substituting the values into the equation.
Don't over-complicate it!
If the 100 coil spring is capable of stretching 0.210 m, then a spring of half this length would be able to stretch 0.210/2 or 0.105 m.
Literally no equations are even required!