Three glasses are placed by a waiter on a light-weight tray. The first glass has
ID: 1446690 • Letter: T
Question
Three glasses are placed by a waiter on a light-weight tray. The first glass has a mass of M_1 = 650 g and is located R_1 = 13 cm from the center of the tray at an angle theta_1 =45 degrees above the positive x-axis. The second glass has a mass of M_2 = 375 g and is located R_2 = 21 cm from the center of the tray at an angle theta_2= 45 degrees below the positive x-axis. The third glass has a mass of M_3 = 325 g and is located R_3 = 16 cm from the center of the tray at an angle theta_3 = 55 degrees above the negative x-axis. A fourth glass of mass M_4 = 875 g is to be placed on the tray so that the center of mass is located at the center of the tray. Randomized Variables M_1 = 650 g R_1 = 13 cm theta_1 = 45 degrees M_2 = 375 g R_2 = 21 cm theta_2 = 45 degrees M_3 = 325 g R_3 = 16 cm theta_3= 55 degrees M_4 = 875 g Write a symbolic equation for the horizontal position from the central x-axis that the fourth glass must be placed so that the horizontal center of massExplanation / Answer
The glasses are located on the tray in terms of polar coordinates. These must
be transformed into rectangular coordinates, which then become the x and y
components of each glass' position vector.
But before that, all angles should be converted to angles that are measured
counterclockwise (CCW) from the positive x-axis.
a)
Angle of M1
= 1
Angle of M2
= 360° - 2
Angle of M3
= 180° - 3
Angle of M4
= to be determined
The rectangular coordinates (x , y) and polar coordinates (R , t) are
related as follows.
y = R sin t and x = R cos t
R^2 = x^2 + y^2 and tan t = y / x
The rectangular coordinates of each glass are as follows
M1
x1 = R1(cos 1) = R1(cos )
y1 = R1(sin 1) = R1(sin )
M2
x2 = R2(cos 2) = R2(cos ) = R2[cos (360° - 2)]
y2 = R2(sin 2) = R2(sin ) = R2[sin (360° - 2)]
M3
x3 = R3(cos 3) = R3(cos ) = R3[cos (180° - 3)]
y3 = R3(sin 3) = R3(sin ) = R3[sin (180° - 3)]
M4
x4 = to be determined
y4 = to be determined
In the x-direction, each x-coordinate is multiplied by its corresponding mass.
The products are then added together. This sum is then divided by the sum of
the masses. This yields the x-coordinate of the center of mass of the aggregate
of M1 + M2 + M3 + M4.
X = x-coordinate of the center of mass of all of the glasses
X = (M1x1 + M2x2 + M3x3 + M4x4) / (M1 + M2 + M3 + M4)
Set X = 1 and solve for x4. This will yield the x-coordinate of M4.
c)
In the y-direction, each y-coordinate is multiplied by its corresponding mass.
The products are then added together. This sum is then divided by the sum of
the masses. This yields the y-coordinate of the center of mass of the aggregate
of M1 + M2 + M3 + M4.
Y = y-coordinate of the center of mass of all of the glasses
Y = (M1y1 + M2y2 + M3y3 + M4y4) / (M1 + M2 + M3 + M4)
Set Y = 1 and solve for y4. This will yield the y-coordinate of M4.