A plastic spring with spring constant 450 N/m has a relaxed length of 0.100m. The spring is positioned vertically on a table, and a charged plastic 1.20-kg sphere is placed on the top end of the spring. Another charged object is suspended above the sphere without making contact. If the length of the spring is now 0.0950m, what are the magnitude and direction of the electric force exerted on the sphere?
To solve this problem, we need to consider both the gravitational force acting on the sphere and the electric force exerted on the sphere.
1. Gravitational force:
The gravitational force on an object is given by the formula: Fg = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of the sphere is 1.20 kg, the gravitational force is: Fg = (1.20 kg) × (9.8 m/s^2) = 11.76 N.
2. Spring force:
The spring force is given by Hooke's Law: Fs = k * x, where k is the spring constant and x is the displacement from the relaxed length.
Given that the spring constant is 450 N/m and the displacement is (0.100m - 0.0950m) = 0.0050m, the spring force is: Fs = (450 N/m) × (0.0050m) = 2.25 N.
3. Total force:
The total force on the sphere is the vector sum of the gravitational force and the spring force.
Since the spring is positioned vertically upward, the gravitational force is acting downward, and the spring force is acting upward.
The magnitude of the total force is given by: Ftotal = |Fg| + |Fs| = 11.76 N + 2.25 N = 14.01 N.
4. Direction of the total force:
Since the spring force is acting upward and the gravitational force is acting downward, the total force will be the difference between these two forces.
Since the spring force is smaller than the gravitational force, the net force will be downward.
Therefore, the direction of the total force is downward.
Therefore, the magnitude of the electric force exerted on the sphere is 14.01 N, and the direction is downward.
To find the magnitude and direction of the electric force exerted on the sphere, we first need to calculate the change in length of the spring ΔL.
Given:
Spring constant, k = 450 N/m
Relaxed length, L₀ = 0.100 m
Final length, L = 0.0950 m
ΔL = L - L₀
= 0.0950 m - 0.100 m
= -0.0050 m
The negative sign indicates that the spring is compressed.
Next, we can use Hooke's Law to find the force exerted by the spring:
F = k * ΔL
F = 450 N/m * (-0.0050 m) [Substituting the values]
F = -2.25 N
The negative sign indicates that the force is acting in the opposite direction of the displacement (compression in this case).
Now, let's consider the gravitational force acting on the sphere:
m = 1.20 kg [Mass of the sphere]
g = 9.8 m/s² [Acceleration due to gravity]
F_gravity = m * g
= 1.20 kg * 9.8 m/s²
= 11.76 N
Since the magnitude of the spring force (-2.25 N) is smaller than the magnitude of the gravitational force (11.76 N), the net force acting on the sphere is the difference between the two forces:
F_net = F_gravity - F
= 11.76 N - (-2.25 N)
= 13.01 N
Therefore, the magnitude of the electric force exerted on the sphere is 13.01 N.
The direction of the electric force is upward, opposite to the direction of gravitational force, since the spring is compressed and trying to restore its original length.