Gravitational force between any two masses is given by
[tex]F = \frac{Gm_1m_2}{r^2}[/tex]
[tex]F = \frac{6.67*10^-11 * 50 * 50}{0.30^2}[/tex]
[tex]F = 1.853 * 10^{-6} N[/tex]
Now net force on it due to two masses is given by
[tex]F_{net} = 2Fcos(\frac{\theta}{2})[/tex]
here [tex]\theta = 60 degree[/tex]
[tex]F_{net} = 2*1.853 * 10^{-6}cos(30)[/tex]
[tex]F_{net} = 3.21 * 10^{-6} N[/tex]
Now acceleration is given by
F= ma
[tex]a = \frac{3.21*10^{-6}}{50}[/tex]
[tex]a = 6.42*10^{-8} m/s^2[/tex]
If one of the masses is released, its initial acceleration is about 6.4 × 10⁻⁸ Newton
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Newton's gravitational law states that the force of attraction between two objects can be formulated as follows:
[tex]\large {\boxed {F = G \frac{m_1 ~ m_2}{R^2}} }[/tex]
F = Gravitational Force ( Newton )
G = Gravitational Constant ( 6.67 × 10⁻¹¹ Nm² / kg² )
m = Object's Mass ( kg )
R = Distance Between Objects ( m )
Let us now tackle the problem !
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Given:
mass of the object = m = 50 kg
distance between the object = R = 0.30 m
Asked:
initial acceleration = a = ?
Solution:
Firstly , let's find the gravitational force between 2 objects as follows:
[tex]F = G \frac{m_1m_2}{R^2}[/tex]
[tex]F = 6.67 \times 10^{-11} \times \frac{ 50 (50)}{0.30^2}[/tex]
[tex]F \approx 1.85 \times 10^{-6} \texttt{ Newton}[/tex]
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Next, we will calculate the resultant vector of these gravitational forces acting on a object:
[tex](\Sigma F)^2 = (F_1)^2 + (F_2)^2 + 2F_1F_2\cos \theta[/tex]
[tex](\Sigma F)^2 = F^2 + F^2 + 2F(F) \cos 60^o[/tex]
[tex](\Sigma F)^2 = 2F^2 + 2F^2(\frac{1}{2})[/tex]
[tex](\Sigma F)^2 = 3F^2[/tex]
[tex]\Sigma F = \sqrt{3F^2}[/tex]
[tex]\Sigma F = F\sqrt{3}[/tex]
[tex]\Sigma F \approx 1.85 \times 10^{-6} (\sqrt{3})[/tex]
[tex]\Sigma F \approx 3.2 \times 10^{-6} \texttt{ Newton}[/tex]
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Finally, we could calculate the initial acceleration of the object by using 2nd-Newton's Law of Motion as follows:
[tex]a = \frac{\Sigma F}{m}[/tex]
[tex]a \approx \frac{3.2 \times 10^{-6}}{50}[/tex]
[tex]a \approx 6.4 \times 10^{-8} \texttt{ m/s}^2[/tex]
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Grade: High School
Subject: Physics
Chapter: Gravitational Fields
