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?
The force exerted by the spring on the sphere can be calculated using Hooke's Law:
F = kΔx
Where F is the force, k is the spring constant, and Δx is the change in length of the spring.
In this case, the change in length is given as:
Δx = 0.0950 m - 0.100 m = -0.005 m
Since the spring is compressed in this case, the force will act in the opposite direction. Therefore, the negative sign is added to the equation.
F = -kΔx
F = -(450 N/m)(-0.005 m)
F = 2.25 N
The magnitude of the force exerted on the sphere is 2.25 N.
The direction of the force is upward since the spring is compressed and wants to return to its relaxed length.
Therefore, the electric force exerted on the sphere is 2.25 N upward.
To solve this problem, we can use Hooke's law to find the change in length of the spring and then relate it to the electric force exerted on the sphere.
Step 1: Calculate the change in length of the spring.
The change in length (∆L) of the spring can be calculated using the formula:
∆L = L - L0
where L is the final length of the spring and L0 is the relaxed length of the spring.
Given:
L0 = 0.100 m
L = 0.0950 m
∆L = 0.0950 m - 0.100 m
∆L = -0.0050 m
Step 2: Calculate the force exerted by the spring.
The force exerted by the spring (∆F) can be calculated using Hooke's law:
∆F = -k∆L
where k is the spring constant.
Given:
k = 450 N/m
∆L = -0.0050 m
∆F = -450 N/m * (-0.0050 m)
∆F = 2.25 N
Since the force is negative, it means that the force is acting in the opposite direction of the displacement of the spring.
Step 3: Calculate the electric force exerted on the sphere.
The electric force (Fe) exerted on the sphere can be calculated using the formula:
Fe = mg
where m is the mass of the sphere and g is the acceleration due to gravity.
Given:
m = 1.20 kg
g = 9.8 m/s^2
Fe = (1.20 kg)(9.8 m/s^2)
Fe = 11.76 N
Step 4: Determine the magnitude and direction of the electric force.
The magnitude of the electric force exerted on the sphere is 11.76 N.
Since the force exerted by the spring is in the opposite direction to the displacement, the electric force will be equal in magnitude but opposite in direction to the force exerted by the spring.
Therefore, the magnitude of the electric force exerted on the sphere is 11.76 N, and its direction is opposite to the force exerted by the spring.