Imagine a spring floating in space. This spring has a very small length when it is unstretched. The spring constant for this spring is 4.2 N/m. Now place 2.6 μC charges on each end of the spring, and allow it to stretch until it reaches a new equilibrium.
What is the new length of the spring?
To determine the new length of the spring, we need to use Hooke's law and the equation for the electrostatic force between charged particles.
1. Start by finding the force exerted by the charges on the spring. The electrostatic force between two point charges can be calculated using Coulomb's law:
F = k * (q1 * q2) / r^2
Where:
- F is the force between the charges
- k is Coulomb's constant (9 × 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges (2.6 μC = 2.6 × 10^-6 C in this case)
- r is the distance between the charges (initial length of the spring when unstretched)
2. Once we have the force, we can use Hooke's law to find the new length of the spring. Hooke's law relates the force F exerted on a spring to the displacement x from its equilibrium position and the spring constant k:
F = k * x
Rearranging the equation, we get:
x = F / k
3. Substitute the values of the force and spring constant into the equation to find the displacement x.
4. Finally, add the displacement x to the initial length of the spring to obtain the new length of the spring.
Please provide the unstretched length (r) of the spring, and I will calculate the new length for you.
To determine the new length of the spring, we need to consider the electrostatic force acting between the charges at each end of the spring.
The force between two charges can be calculated using Coulomb's law:
F = k * |q₁ * q₂| / r²
Where:
F is the electrostatic force between the charges,
k is the electrostatic constant (approximately 9 x 10^9 Nm²/C²),
q₁ and q₂ are the magnitudes of the charges (in this case, 2.6 μC, which is 2.6 x 10^(-6) C),
and r is the distance between the charges (which is equal to the length of the spring).
Since the spring is in equilibrium, the electrostatic force must be balanced by the spring force (F = k * x). Here, x is the displacement of the spring from its unstretched position.
Setting these two forces equal to each other, we have:
k * x = k * |q₁ * q₂| / r²
Simplifying, we have:
x = |q₁ * q₂| / (r² * spring constant)
Substituting the given values, we have:
x = |2.6 x 10^(-6) C * 2.6 x 10^(-6) C| / (r² * 4.2 N/m)
Calculating the right-hand side of the equation, we obtain:
x = 6.76 x 10^(-12) C² / (r² * 4.2 N/m)
To find the new length of the spring, we substitute the known values and solve for r:
r² = 6.76 x 10^(-12) C² / (x * 4.2 N/m)
r² = 1.61 x 10^(-12) C² / (x N/m)
Taking the square root of both sides:
r = sqrt(1.61 x 10^(-12) C² / (x N/m))
Calculating, we find:
r ≈ 1.27 x 10^(-6) m
Therefore, the new length of the spring when it reaches equilibrium is approximately 1.27 micrometers (μm).