A small sphere of charge q1 = 0.840 µC hangs from the end of a spring as in Figure a. When another small sphere of charge q2 = -0.546 µC is held beneath the first sphere as in Figure b, the spring stretches by d = 3.18 cm from its original length and reaches a new equilibrium position with a separation between the charges of r = 4.95 cm. What is the force constant of the spring?
Use Hooke's Law: |F| = k |d|
Use Coulomb's Law: |F| = ke |q1 q2| / r^2
Equate forces and rearrange.
k = ke |q1 q2|/(|d| r^2)
Where:
q1 = 0.840×10^(-6) C
q2 = -0.546×10^(-6) C
d = 3.18×10^(-2) m
r = 4.95×10^(-2) m
ke = 8.9875517873681764×10^9 N·m^2/C^2 (Coulomb's Constant)
To find the force constant of the spring, we can use Hooke's Law which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position.
1. First, let's calculate the force exerted by the spring on the system when the spring is stretched by a distance of d.
F = k * x, where F is the force, k is the force constant, and x is the displacement.
In this case, x = d = 3.18 cm = 0.0318 m.
2. Now, we need to calculate the net electrostatic force between the two charged spheres.
F_electric = k_e * |q1 * q2| / r^2, where F_electric is the electrostatic force, k_e is the electrostatic constant (k_e = 8.99 × 10^9 Nm^2/C^2), q1 and q2 are the charges, and r is the separation between the charges.
In this case, q1 = 0.840 µC = 0.840 × 10^-6 C, q2 = -0.546 µC = -0.546 × 10^-6 C, and r = 4.95 cm = 0.0495 m.
3. Since the system is in equilibrium, the electrostatic force is balanced by the force exerted by the spring.
F = F_electric, so we can set the two equations equal to each other.
k * d = k_e * |q1 * q2| / r^2
4. Rearrange the equation to solve for the force constant k.
k = (k_e * |q1 * q2|) / (d * r^2)
5. Substitute the known values into the equation and calculate the force constant k.
k = (8.99 × 10^9 Nm^2/C^2 * |0.840 × 10^-6 C * -0.546 × 10^-6 C|) / (0.0318 m * (0.0495 m)^2)
Simplifying the equation should give you the force constant of the spring in N/m.
To find the force constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Let's denote the force constant of the spring as k. The displacement of the spring is given as d = 3.18 cm, and we need to find the value of k.
According to Hooke's Law, the force exerted by the spring is given by the equation:
F = -k * d
where F is the force, k is the force constant, and d is the displacement.
In this case, the displacement of the spring is in the opposite direction of the force, so we use a negative sign.
First, let's convert the displacement from centimeters to meters:
d = 3.18 cm = 0.0318 m
Next, we need to find the force exerted by the spring. The force between the two charged spheres can be calculated using Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
where F is the force, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), q1 and q2 are the charges of the spheres, and r is the separation between the charges.
Substituting the given values:
F = k * (|0.840 µC| * |(-0.546 µC)|) / (0.0495 m)^2
The magnitudes of the charges can be used because electrostatic forces are scalar quantities.
F = k * (0.840 × 10^-6 C) * (0.546 × 10^-6 C) / (0.0495 m)^2
Now, we need to equate this force to the force exerted by the spring:
-k * d = k * (0.840 × 10^-6 C) * (0.546 × 10^-6 C) / (0.0495 m)^2
Since the displacements cancel out, we can solve for the force constant:
k = [(0.840 × 10^-6 C) * (0.546 × 10^-6 C) / (0.0495 m)^2] / d
Now, we can substitute the given values:
k = [(0.840 × 10^-6 C) * (0.546 × 10^-6 C) / (0.0495 m)^2] / 0.0318 m
Calculating this expression will give us the force constant of the spring.