A plastic spring with spring constant 450 N/m has a relaxed length of 0.100 m. 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 0.0950 m, what are the magnitude and direction of the electric force exerted on the sphere?
To find the electric force exerted on the sphere, we need to consider the forces acting on it: the force due to gravity and the force due to the compressed spring.
1. Force due to gravity: The weight of the sphere can be calculated using the formula:
F_gravity = mass × acceleration due to gravity
F_gravity = 1.20 kg × 9.8 m/s^2
F_gravity = 11.76 N
This force acts downward.
2. Force due to the compressed spring: The force exerted by a spring can be calculated using Hooke's law:
F_spring = -k × Δx
where k is the spring constant and Δx is the change in length of the spring.
Δx = Final length - Relaxed length
= 0.0950 m - 0.100 m
= -0.005 m
Note: The negative sign indicates that the spring is compressed.
F_spring = -450 N/m × (-0.005 m)
= 2.25 N
This force acts upward.
Now, let's consider the net force acting on the sphere:
Net force = F_spring + F_gravity
= 2.25 N + 11.76 N
= 14.01 N
The magnitude of the electric force exerted on the sphere is 14.01 N.
Since the net force is directed upward, the direction of the electric force exerted on the sphere is also upward.
To find the magnitude and direction of the electric force exerted on the sphere, we need to consider the gravitational force acting on the sphere and the net force acting on the spring.
Given:
Spring constant k = 450 N/m
Relaxed length L₀ = 0.100 m
Final length L = 0.0950 m
Sphere mass m = 1.20 kg
First, let's calculate the change in length of the spring (ΔL):
ΔL = L - L₀
ΔL = 0.0950 m - 0.100 m
ΔL = -0.0050 m
Now, let's calculate the gravitational force acting on the sphere (F_grav):
F_grav = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
F_grav = 1.20 kg * 9.8 m/s²
F_grav ≈ 11.76 N
Next, let's calculate the spring force (F_spring) using Hooke's Law:
F_spring = k * ΔL
F_spring = 450 N/m * (-0.0050 m)
F_spring ≈ -2.25 N
The negative sign indicates that the spring is being compressed.
Finally, let's calculate the net force acting on the sphere (F_net):
F_net = F_spring + F_grav
F_net = -2.25 N + 11.76 N
F_net ≈ 9.51 N
Therefore, the magnitude of the electric force exerted on the sphere is approximately 9.51 N. The direction of the electric force is not specified in the given information and would depend on the charge and position of the other object suspended above the sphere.