Question

An 8.75 kg point mass and a 14.0 kg point mass are held in place 50.0 cm apart. A particle of mass m is released from a point between the two masses 17.0 cm from the 8.75 kg mass along the lines connecting the two fixed masses.
A. Find the magnitude (a) of the acceleration of the particle in m/s^2.
B. Find the direction of the acceleration of the particle. Either (i) The acceleration of the particle is toward the 8.75kg mass, or (ii) The acceleration of the particle is toward the 14.0kg mass.

Answer 1

Answer

given,

mass of the = m₁ = 8.75 Kg

another mass of the object = m₂ = 14 Kg

distance between them = 50 cm

R₁ = 17 cm

R₂ = 50 -17 = 33 cm

a) Force applied due to the Mass 8.75  in +ve x- direction

$F_1 = \dfrac{GM_1 m}{R_1^2}$

$F_1 = \dfrac{6.67\times 10^{-11} \times 8.75\ m}{0.17^2}$

$F_1 = 2.019\times 10^{8}\ m$

Force applied due to mass 14 Kg in -ve x-direction

$F_2 = \dfrac{GM_2 m}{R_2^2}$

$F_2 = \dfrac{6.67\times 10^{-11} \times 14\ m}{0.33^2}$

$F_2 = 0.857\times 10^{8}\ m$

net force

F = F₁ + F₂

$F = 2.019\times 10^{8}\ m - 0.857\times 10^{8}\ m$

$F = 1.162\times 10^{8}\ m$

Using newton second law

$a = \dfrac{F}{m}$

$a = \dfrac{ 1.162\times 10^{8}\ m}{m}$

$a =1.162\times 10^{8} \ m/s^2$

b) As the acceleration of mass comes out to be  +ve hence, the direction will be toward the mass of 8.75 Kg

Answer 2

(a) The acceleration of the particle is 1.163 x 10⁻⁸ m/s².

(b) The acceleration of the particle is toward the 8.75kg mass.

Force between the masses

The force between the masses is determined by applying Newton's law of universal gravitation as shown below;

F = Gm₁m₂/R²

Force between 8.75 kg and mass, m

$F_1 = \frac{6.67 \times 10^{-11} \times 8.75 \times m}{(0.17)^2} \\\\F_1 = 2.02 \times 10^{-8} m$

Force between 14 kg and mass, m

distance between 14 kg and m = 50 cm - 17 cm = 33 cm

$F_2 = \frac{6.67 \times 10^{-11} \times 14 \times m}{(0.33)^2} \\\\F_2 = 8.57\times 10^{-9} m= 0.857\times 10^{-8}m$

Net force

The net force on the masses;

F(net) = F1 + F2

F(net) = 2.02 x 10⁻⁸m - 0.857 x 10⁻⁸m = 1.163 x 10⁻⁸ m

Acceleration of the particle

F = ma

a = F/m

a = (1.163 x 10⁻⁸ m)/m

a = 1.163 x 10⁻⁸ m/s²

Since the acceleration if positive it will be directed towards 8.75 kg mass.

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