A massless spring is attached to a support at one end and has a 2.0 μC charge glued to the other end. A -4.0 μC charge is slowly brought near. The spring has stretched 1.2 cm when the charges are 1.5 cm apart.
What is the spring constant of the spring?
k = ?
To find the spring constant (k), we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
The formula for Hooke's Law is: F = -kx
Where:
F = Force exerted by the spring
k = Spring constant
x = Displacement from the equilibrium position
In this case, the displacement (x) is 1.2 cm, and the force (F) is the force exerted by the spring.
However, we need to consider the forces due to both charges. The force between two point charges can be calculated using Coulomb's Law:
F = k_e * |q1 * q2| / r^2
Where:
F = Force between the charges
k_e = Coulomb's constant (8.99 x 10^9 N*m^2/C^2)
q1 and q2 = Charges of the two objects
r = Distance between the charges
In this case, q1 = 2.0 μC and q2 = -4.0 μC, and the distance (r) is 1.5 cm.
First, let's convert the charges to Coulombs:
2.0 μC = 2.0 x 10^-6 C
-4.0 μC = -4.0 x 10^-6 C
Next, let's convert the distance to meters:
1.5 cm = 1.5 x 10^-2 m
Now, let's calculate the force between the charges using Coulomb's Law:
F = (8.99 x 10^9 N*m^2/C^2) * |(2.0 x 10^-6 C) * (-4.0 x 10^-6 C)| / (1.5 x 10^-2 m)^2
F = (8.99 x 10^9 N*m^2/C^2) * (8.0 x 10^-12 C^2) / (2.25 x 10^-4 m^2)
F = 2.78 N
Since the displacement x is 1.2 cm = 1.2 x 10^-2 m, and the force F is 2.78 N, we can use Hooke's Law to find the spring constant (k):
2.78 N = -k * (1.2 x 10^-2 m)
Let's solve for k:
k = -2.78 N / (1.2 x 10^-2 m)
k ≈ -231.7 N/m (Approximately)
To find the spring constant (k) of the massless spring, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement produced:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
In this scenario, we are given the displacement (x) as 1.2 cm, and we need to find the spring constant (k). However, we don't have the force (F) directly, so we need to find it using the formula for the electric force between two charged objects:
F = k_e * (|q1 * q2| / r^2)
Where:
F is the electric force between the charges,
k_e is the electrostatic constant (8.99 x 10^9 Nm^2/C^2),
q1 and q2 are the charges,
and r is the distance between the charges.
Here, q1 is 2.0 μC and q2 is -4.0 μC, which we convert to Coulombs as 2.0 x 10^-6 C and -4.0 x 10^-6 C, respectively. The distance (r) is given as 1.5 cm, which we convert to meters as 1.5 x 10^-2 m.
Now, we can equate the electric force with the force from the spring:
F = -kx
k_e * (|q1 * q2| / r^2) = k * x
Substituting the given values:
8.99 x 10^9 Nm^2/C^2 * (|2.0 x 10^-6 C * -4.0 x 10^-6 C| / (1.5 x 10^-2 m)^2) = k * 1.2 x 10^-2 m
Now, we can solve for k:
k = (8.99 x 10^9 Nm^2/C^2 * (2.0 x 10^-6 C * 4.0 x 10^-6 C) / (1.5 x 10^-2 m)^2) / (1.2 x 10^-2 m)
k ≈ 7.1 N/m
Therefore, the spring constant (k) is approximately 7.1 N/m.