Question

Objects with masses of 255 kg and a 555 kg are separated by 0.390 m.

Answer 1

Answer:

(a). The net gravitational force is $4.20\times10^{-6}\ N$

(b). The position is at 0.232 m.

Explanation:

Given that,

Mass of one object M = 255 kg

Mass of another object M'= 555 kg

Separation = 0.390 m

(a). We need to calculate the net gravitational force

Using formula of force

$F_{net}=\dfrac{GM'm}{r^2}-\dfrac{GMm}{r^2}$

$F_{net}=\dfrac{Gm(M'-M)}{r^2}$

Put the value into the formula

$F_{net}=\dfrac{6.67\times10^{-11}\times32.0(555-255)}{(0.390)^2}$

$F_{net}=4.20\times10^{-6}\ N$

The net gravitational force is $4.20\times10^{-6}\ N$

(b). We need to calculate the position

Force from 555 mass = Force from 255 mass

$\dfrac{Gm\times555}{x^2}=\dfrac{Gm\times255}{(0.390-x)^2}$

$555(0.390-x)^2=255x^2$

$300x^2-0.78x+84.4155=0$

$x=0.232\ m$

The position is at 0.232 m.

Hence, (a). The net gravitational force is $4.20\times10^{-6}\ N$

(b). The position is at 0.232 m.

Answer 2

Answer:

a. the net force F = 1.684 × 10⁻⁵ N in the direction of the 555 kg mass. b. x =  0.1576 m or -0.8205 m

Explanation:

a. The force of gravitational attraction between two masses m₁ and m₂ separated by a distance, r is given by F = Gm₁m₂/r².

Let m₁, m₂ and m₃ be the masses and m₂ is midway between m₁ and m₃.

The force of attraction between m₁ and m₂ is F₁ = Gm₁m₂/r² and the force of attraction between m₂ and m₃ is F₂ = Gm₂m₃/r². The net force is F₁ - F₂ =  Gm₁m₂/r² - Gm₂m₃/r²= Gm₂/r²(m₁ - m₃).

m₁ = 255 kg, m₂ = 555 kg and r = 0.390m/2 = 0.195 m

F = 6.67 × 10⁻¹¹ Nm²/kg² × 32 kg ( 255 - 555) kg/ (0.195 m)²

  = 213.44 × 10⁻¹¹ (- 300)/(0.038025)N

  = -1683944.77 × 10⁻¹¹ N

  = -1.684 × 10⁻⁵ N.

So, the net force F = 1.684 × 10⁻⁵ N in the direction of the 555 kg mass.

b. Let x be the distance from the 255 kg mass and d - x be the distance from the 555 kg mass at zero net force, then

F = Gm₁m₂/x² = Gm₂m₃/(d - x)²

Simplifying,

(d - x)²/x² = Gm₂m₃/Gm₁m₂ = m₃/m₁.

Taking square root of both sides, we have

(d - x)/x = ± √ (m₃/m₁)

d/x - 1 = ± √ (m₃/m₁)

d/x = 1 ± √ (m₃/m₁)

x = d/(1 ± √ (m₃/m₁)) = 0.390/(1  ± √ (555/255)) = 0.390/(1  ± √ (2.1765)) = 0.390/(1  ± 1.4753)

x = 0.390/2.4753 or 0.390/-0.4753 = 0.1576 m or -0.8205 m.

x = 0.1576 m ≅ 0.158 m and -0.8205 m ≅ -0.821 m

x = 0.158 m or -0.821 m