Question

Two spheres A and B of mass 7.5 kg and 8.3 kg respectively are separated by a distance of 0.56 m.

(a) Calculate the magnitude of the gravitational force A exerts on B and B exerts on A.
force A exerts on B N
force B exerts on A N
(b) If the force between the spheres is now 2.60 10-9 N, how far apart are their centers? m

Answer 1

(a) The magnitude of the gravitational force A exerts on B is approximately 6.17 ×
`10^(-8)` N, and the force B exerts on A is also 6.17 × 10^-8 N.

(b) If the force between the spheres is 2.60 ×
`10^(-9)` N, their centers are approximately 0.317 m apart.

Step-by-step explanation:

In part (a), we calculate the gravitational force between the spheres using Newton's law of universal gravitation:

`\[ F = \frac{{G \cdot m_A \cdot m_B}}{{r^2}} \]`

where
`\( G \)` is the gravitational constant

`(\(6.674 * 10^(-11) \, \text{Nm}^2/\text{kg}^2\)), \( m_A \) and \( m_B \)` are the masses of the spheres, and
`\( r \)` is the separation distance.

For the force A exerts on B:

`\[ F_(AB) = \frac{{G \cdot m_A \cdot m_B}}{{r^2}} = \frac{{(6.674 * 10^(-11)) \cdot (7.5) \cdot (8.3)}}{{(0.56)^2}} \]`

Similarly, for the force B exerts on A:

`\[ F_(BA) = \frac{{G \cdot m_A \cdot m_B}}{{r^2}} = \frac{{(6.674 * 10^(-11)) \cdot (7.5) \cdot (8.3)}}{{(0.56)^2}} \]`

In part (b), if the force between the spheres is
`\(2.60 * 10^(-9) \, \text{N}\),` we rearrange the formula to solve for
`\( r \):`

`\[ r = \sqrt{\frac{{G \cdot m_A \cdot m_B}}{{F}}} \]`

Substituting the given values:

`\[ r = \sqrt{\frac{{(6.674 * 10^(-11)) \cdot (7.5) \cdot (8.3)}}{{2.60 * 10^(-9)}}} \]`

This gives us the separation distance
`\( r \),` which is approximately 0.317 m.

Answer 2

The magnitude of the gravitational force A exerts on B is approximately 7.01x10⁻⁷ N and the force B exerts on A is also 7.01x10⁻⁷ N. When the force between the spheres is 2.60x10⁻⁹ N, their centers are approximately 0.521 m apart.

Step-by-step explanation:

To calculate the magnitude of the gravitational force between two spheres, we can use Newton's Law of Universal Gravitation. According to this law, the gravitational force between two objects is given by:

F = G ×(m1 ×m2) / r²

Where F is the magnitude of the gravitational force, G is the gravitational constant (6.67 × 10⁻¹¹ Nm²/kg²), m1 and m2 are the masses of the two spheres, and r is the distance between their centers. Using this equation, we can calculate:

(a) Force A exerts on B:

Substituting the values, F = (6.67 × 10⁻¹¹ Nm²/kg²) ×(7.5 kg) ×(8.3 kg) / (0.56 m)²

Calculating this, we get a magnitude of approximately 7.01x10⁻⁷ N.

(b) Force B exerts on A:

The gravitational force between A and B is attractive, so the magnitude will be the same as force A exerts on B, which is 7.01x10⁻⁷ N.

To calculate the distance between the spheres when the force is 2.60x10⁻⁹ N, we can rearrange the equation as:

r = √((G ×(m1 × m2)) / F)

Substituting the values, r = √((6.67 × 10⁻¹¹ Nm²/kg²) ×(7.5 kg) ×(8.3 kg) / (2.60x10⁻⁹ N))

Calculating this, we get a distance of approximately 0.521 m.