7. (40 points) Three uniform spheres are fixed at the three corners of a square
ID: 1795273 • Letter: 7
Question
7. (40 points) Three uniform spheres are fixed at the three corners of a square of side a = 0.50 m. A 4th sphere, of mass m "represented by the dot, is placed at the lower RIGHT corner of square-at point P. Note: The sphere in the upper LEFT corner has mass M = 2.00 kg and the spheres in the LOWER left (AT ORIGIN 0) and UPPER right corners have equal masses m = 1.00 kg. The sphere in the lower RIGHT corner has mass m' 0.0150 kg. See left diagram in the figure below. (a) (8 points) What is the magnitude of the force on the sphere of mass m' due to the sphere directly above it in the upper RIGHT corner. (b) (8 points) What is the magnitude of the force on the sphere of mass m' due to the sphere across the diagonal in the upper LEFT corner? (c) (8 point) What is the magnitude of the force on the sphere of mass m' due to the sphere directly to the LEFT of it in the lower LEFT corner? (d) (8) What is the magnitude of the resultant force on the sphere of mass m'? (e) (8) What is the direction of the resultant force on the sphere of mass m'? Show this angle by drawing the resultant on the blank x-y axes in the correct quadrant. Label the angle the vector makes with the x-axis.Explanation / Answer
m' = 0.0150 kg
M = 2.0 kg
m = 1 kg
a = 0.50 m
a)
The magnitude of the force on the sphere of mass m’ due to the sphere directly above it in the upper right corner is
F = Gm'*m/a^2 = 6.754 * 10^-11 * 0.0150 * 1.0 / 0.5^2 = 4.00 *10^ -12 N .
b) Similarly Force on m' due to the sphere across the diagonal .
F = Gm'*M/(2a^2) = 4.00 * 10^ -12 N .
c) The magnitude of the force on the sphere of mass m’ due to the sphere directly to the left of it in the lower left corner
F = Gm'*m/a^2 = 6.754 * 10^-11 * 0.0150 * 1.0 / 0.5^2 = 4.00 * 10^ -12 N .
d)
X-component force = 4.00 * 10 ^-12 N + 4.00 * 10^ -12 cos (135) N .
Y-component force = 4.00 * 10 ^-12 N + 4.00 * 10^ -12 sin (135) N .
Using the Pythagorean Theorem:
The Magnitude of the resultant force on the sphere of mass m' = 1.2 * 10^ -11 N.
e)
The force makes an angle of 135 degrees with the positive X-axis.
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