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 3.6 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 N y component N (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 m/s2 y component m/s2

Explanation / Answer

calculate the total force on m as the vector sum of individual objects on m. The magnitude of each is: F = G M m / R^2 = 6.673e-11x 3 x 2 / (0.1)^2 = 4.0038e-8 newtons so if you draw 5 arrows of equal length on a graph, from m toward each of the five objects, and then arrange the arrows as a string, connecting head-to-tail in sequence without changing their direction, what do you get? the distance travelled toward the middle will be 1+sqrt(2) times the unit length. this means, (1+sqrt(2)) x 4.0038e-8 = 9.666e-8 2. if you use the acceleration for this, instead of force, then you can plug in an infinitessimally small mass m into the equation: since F = ma, the equation above becomes F = ma = G M m / R^2 --> a = G M / R^2.

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