Answer :
The magnitude of the combined gravitational field at the point (8.00 m along the positive x-axis) is 4.67 × 10⁻⁷ N/kg .
To find the combined gravitational field at a point due to two masses, we can treat each mass individually and then add the vector contributions from each. Here's how we can solve this problem:
Step 1: Define the masses and positions:
Mass of each object (m1 and m2): 1.00 × 10³ kg
Position of the first mass (x1): 0.00 m (origin)
Position of the second mass (x2): 4.00 m
Point of interest (x3): 8.00 m
Step 2: Calculate the gravitational constant (G):
G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻²
Step 3: Calculate the vectors from each mass to the point of interest:
Vector from the first mass (r1): [x3 - x1, 0, 0] = [8.00 m, 0, 0]
Vector from the second mass (r2): [x3 - x2, 0, 0] = [4.00 m, 0, 0]
Step 4: Calculate the magnitudes of the vectors:
r1_mag = ||r1|| = √(8.00² + 0² + 0²) = 8.00 m
r2_mag = ||r2|| = √(4.00² + 0² + 0²) = 4.00 m
Step 5: Calculate the gravitational force due to each mass:
Force due to the first mass (F1):
F1 = G * m1 * m2 / (r1_mag³ * r1)
= (6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻²) * (1.00 × 10³ kg) * (1.00 × 10³ kg) / (8.00³ m³ * [8.00, 0, 0])
= [−3.33714844 × 10⁻⁷ N, 0, 0]
Force due to the second mass (F2):
F2 = G * m2 * m2 / (r2_mag³ * r2)
= (6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻²) * (1.00 × 10³ kg) * (1.00 × 10³ kg) / (4.00³ m³ * [4.00, 0, 0])
= [−1.33485937 × 10⁻⁷ N, 0, 0]
Step 6: Combine the forces (vector addition):
Gravitational field at the point (F_total): F_total = F1 + F2 = [-3.33714844 × 10⁻⁷ N - 1.33485937 × 10⁻⁷ N, 0, 0] = [-4.67200781 × 10⁻⁷ N, 0, 0]
Step 7: Calculate the magnitude of the combined gravitational field
Magnitude of F_total: ||F_total|| = √((-4.67200781 × 10⁻⁷ N)² + 0² + 0²) = 4.67200781 × 10⁻⁷ N
Therefore, the magnitude of the combined gravitational field at the point (8.00 m along the positive x-axis) is 4.67 × 10⁻⁷ N/kg .
Final answer:
Explanation on how to calculate the combined gravitational field at a specific point due to two masses.
Explanation:
Gravitational field is a vector quantity that describes the force experienced by a unit mass at a certain point in space due to a gravitational source. In this scenario, to find the combined gravitational field at 8.00 m along the positive x-axis, we first calculate the individual gravitational fields of the two masses at that point and then add them vectorially to get the total field.
To solve the problem:
- Calculate the gravitational field at 8.00 m due to the first mass.
- Calculate the gravitational field at 8.00 m due to the second mass.
- Add the two vectors using vector addition to obtain the total gravitational field at 8.00 m.
