Two thin hollow plastic spheres, about the size of a ping-pong ball with masses (m1 = m2 = 2e-3 kg) have been rubbed with wool. Sphere 1 has a charge q1 = -6e-9 C and is at location < 40e-2, -20e-2, 0 > m. Sphere 2 has a charge q2 = -6e-9 C and is at location < -50e-2, 50e-2, 0> m. It will be useful to draw a diagram of the situation, including the relevant vectors.
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What is the magnitude of the gravitational force exerted on q2 by q1?
What is the gravitational force (vector) exerted on q2 by q1?
To find the magnitude of the gravitational force exerted on q2 by q1, we can use the formula for gravitational force:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2), m1 and m2 are the masses of the objects, and r is the distance between the centers of the two objects.
In this case:
m1 = m2 = 2e-3 kg (given)
G = 6.67430 × 10^-11 N(m/kg)^2 (constant)
r = distance between the centers of the two spheres. To find this, we can use the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
where (x1, y1, z1) and (x2, y2, z2) are the respective coordinates of the two spheres.
Given coordinates:
Sphere 1: (x1, y1, z1) = (40e-2, -20e-2, 0) m
Sphere 2: (x2, y2, z2) = (-50e-2, 50e-2, 0) m
Substituting the values into the distance formula:
r = sqrt((-50e-2 - 40e-2)^2 + (50e-2 - (-20e-2))^2 + (0 - 0)^2)
Simplifying:
r = sqrt((-90e-2)^2 + (70e-2)^2)
r = sqrt((8100e-4) + (4900e-4))
r = sqrt(13000e-4)
r = sqrt(130) m
Now we can calculate the magnitude of the gravitational force using the formula mentioned earlier:
F = G * (m1 * m2) / r^2
F = (6.67430 × 10^-11) * (2e-3 * 2e-3) / (sqrt(130))^2
F = (6.67430 × 10^-11) * (4e-6) / 130
F = (26.6972 × 10^-11) / 130
F ≈ 2.05363 × 10^-13 N
Therefore, the magnitude of the gravitational force exerted on q2 by q1 is approximately 2.05363 × 10^-13 N.
To find the gravitational force vector exerted on q2 by q1, we need to consider the direction. Since both charges are negative and located in their respective positions, the gravitational force vector will be attractive and points from Sphere 1 towards Sphere 2. The force vector can be represented as:
F = -2.05363 × 10^-13 N (in the direction from Sphere 1 to Sphere 2)