A spiral spring is compressed so as to add U units of potential energy to it.

When this spring is instead stretched two-thirds of the distance it was compressed, its remaining potential energy in the same units will be

a. 2U/3
b. 4U/9
c. U/3
d. U/9

Please try these yourself before asking

(2/3)^2 = 4/9

The potential energy of a compressed or stretched spring is given by the equation:

Potential energy (PE) = (1/2)kx^2

Where:
- k is the spring constant, which represents the stiffness of the spring
- x is the displacement from the equilibrium position of the spring

In this case, when the spring is compressed by U units, the potential energy added to it is U units. Therefore, we can write:

U = (1/2)kx^2

Solving for k, we get:

k = (2U) / x^2

Now, let's calculate the remaining potential energy when the spring is stretched two-thirds of the distance it was compressed.

Let's assume the original compression distance is x. The stretch distance would then be (2/3)x.

To find the remaining potential energy, we need to calculate the potential energy at this new stretched position.

PE remaining = (1/2)k (2/3x)^2

Substituting the equation we found for k above:

PE remaining = (1/2) * (2U / x^2) * (2/3x)^2

PE remaining = (1/6)U

Therefore, the remaining potential energy in the spring, in the same units, is U/6.

The correct option would be:
c. U/6

To determine the remaining potential energy in the spring when it is stretched two-thirds of the distance it was compressed, we can use the concept of Hooke's Law and the principle of conservation of energy.

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, this can be expressed as F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement.

The potential energy stored in a spring can be calculated using the formula U = (1/2)kx^2, where U is the potential energy, k is the spring constant, and x is the displacement.

Now, let's go step-by-step to find the answer:

1. Let's assume that the original displacement (when the spring was compressed) was x.

2. When the spring is stretched two-thirds of the distance it was compressed, the new displacement will be (2/3)x.

3. The force exerted by the spring at the new displacement can be found using Hooke's Law:

F = -k(2/3)x.

4. The potential energy stored in the spring at the new displacement can be calculated using the formula:

U' = (1/2)k(2/3)x^2.

5. Simplifying the equation:

U' = (1/2)(2/3)x^2.

U' = (1/3)x^2.

Based on the simplified equation, we can see that the remaining potential energy in the spring when it is stretched two-thirds of the distance it was compressed is U/3.

Therefore, the correct answer is option c. U/3.

hi, don't know the answer, i didn't even go to college.... i am trying to invent a device that will aid in handicapped sports... i am trying to load a spring with the least amount of energy and compound it into the highest kinetic force i can.. only thing loading this spring will be a HC person. For instance when leaning left the left spring is loaded as the right spring is extended, then leaning right the opposite occurs..

sorry to bug you