Five objects, each of mass M, are equally spaced on the arc of a semicircle of r

Question

Five objects, each of mass M, are equally spaced on the arc of a semicircle of radius R as shown in the figure below. An object of mass m is located at the center of curvature of the arc. (a) If M is 2.6 kg, m is 2.4 kg, and R is 9.0 cm, what is the gravitational force on the particle of mass m due to the five objects? Choose right to be the +x direction and up to be the +y direction. x component ??? y component ??? (b) If the object whose mass is m is removed, what is the gravitational field at the center of curvature of the arc? Again, choose right to be the +x direction and up to be the +y direction. x component ??? y component ???

Explanation / Answer

Given that mass M = 3.4 kg

mass m = 1.8 kg

R = 9 cm = 0.09 m

a)

For the net gravitational force on m due to five masses is given by:

The horizontal component of the force get cancelled due to symmetry

Fx = 0

The vertical component of the force is given by:

Fy = GMm/R2 (sin0 + sin45 + sin90 + sin 135 + sin180)

Fy =  GMm/R2 (2sin45 + 1)

Fy = (6.67*10^-11 Nm2/kg2)(3.4 kg)(1.8 kg)/0.09 2 (2 + 1)

Fy = 1.216*10-7 N

b)

The gravitational field at the center of curvature of the arc is given by

g = F/m

The X component is zero

gx = 0

The Y component is

gy = 1.216*10-7/ 1.8kg

gy = 6.75*10-8 m/s2

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