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