A spring has a spring stiffness constant, k, of 420 N/m .
1. How much must this spring be stretched to store 30 J of potential energy?
Potential energy = 30 J,
Spring stiffness constant k = 420 N/m
distance x = ?
Potential energy = 1/2*k*x^2
=> 30 = 1/2*420*x^2
=> x = √(30/210)
= 0.378 m
Well, that spring must be feeling pretty "tension"-al right now! To calculate how much it needs to be stretched, we can use the formula for potential energy stored in a spring, which is PE = (1/2)kx^2.
Since we want to find x (the stretch), we can rearrange the equation to solve for x. Let's do some math magic!
30 J = (1/2)(420 N/m)x^2
Now, let's simplify the equation a bit:
60 J = 420 N/m x^2
Next, let's divide both sides by 420 N/m to get rid of it:
0.143 J/m = x^2
Now, we need to get rid of that pesky square by taking the square root!
x = √(0.143 J/m)
So, the spring needs to be stretched by approximately √(0.143 J/m) meters to store 30 J of potential energy.
To calculate the amount the spring must be stretched to store a certain amount of potential energy, we can use the formula for potential energy stored in a spring:
Potential Energy (PE) = (1/2) * k * x^2,
where:
- PE is the potential energy stored in the spring,
- k is the spring stiffness constant,
- x is the displacement or stretch of the spring.
In this case, we are given that the spring stiffness constant, k, is 420 N/m and the potential energy, PE, is 30 J.
Using the formula and rearranging it to solve for x, we have:
30 J = (1/2) * 420 N/m * x^2.
We can simplify this equation:
60 J = 420 N/m * x^2.
To isolate x^2, divide both sides of the equation by 420 N/m:
(60 J) / (420 N/m) = x^2.
Simplify the right side of the equation:
0.1429 J/m = x^2.
Now, take the square root of both sides to solve for x:
√(0.1429 J/m) = √(x^2).
x ≈ √0.1429 J/m.
Using a calculator, the approximate value for x is:
x ≈ 0.37822 m.
Therefore, the spring must be stretched approximately 0.37822 meters to store 30 J of potential energy.
To find out how much the spring must be stretched, we need to use the formula for potential energy stored in a spring:
Potential energy (PE) = (1/2) * k * x^2
where k is the spring stiffness constant and x is the displacement or stretch of the spring from its equilibrium position.
In this case, we are given the potential energy (PE) as 30 J and the spring constant (k) as 420 N/m. We can rearrange the formula to solve for x:
PE = (1/2) * k * x^2
30 J = (1/2) * 420 N/m * x^2
Divide both sides by (1/2) * 420 N/m:
60 J/m = x^2
Now, to find x, take the square root of both sides:
x = sqrt(60 J/m)
Therefore, the spring must be stretched by sqrt(60 J/m) meters to store 30 J of potential energy.