Question

Three glasses are placed by a waiter on a light-weight tray. The first glass has a mass of M1 = 525 g and is located R1 = 13 cm from the center of the tray at an angle θ1 = 35 degrees above the positive x-axis. The second glass has a mass of M2 = 325 g and is located R2 = 22 cm from the center of the tray at an angle θ2 = 45 degrees below the positive x-axis. The third glass has a mass of M3 = 225 g and is located R3 = 16 cm from the center of the tray at an angle θ3 = 25 degrees above the negative x-axis. A fourth glass of mass M4 = 875 g is to be placed on the tray so that the center of mass is located at the center of the tray. a) 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 mass of the four glasses is at the center of the tray. b) Calculate the numeric value of the horizontal position from the central x-axis of the fourth glass in cm.

Answer 1

To balance the center of mass on the tray, the fourth glass must be placed in a specific position calculated using the masses and positions of the existing glasses. We assume the fourth glass is along the x-axis (angle θ_4 = 0) for simplicity, and calculate its distance as approximately -4 cm from the center of the tray.

Step-by-step explanation:

To calculate the horizontal position from the central x-axis where the fourth glass must be placed, we need to use the concept of center of mass of a system. For the center of mass to be at the center of the tray, the sum of moments (mass times the perpendicular distance from the axis) of all the glasses around the x-axis must equal zero. To find the position of the fourth glass, we use the equation:

Σ (m_i * r_i * cos(θ_i)) = 0

where m is the mass of each glass, r is the distance from the center, and θ is the angle from the horizontal axis. Let's call the horizontal position of the fourth glass R4 and its angle from the positive x-axis θ_4.

Symbolic equation for x-coordinate of the center of mass is:

(M1 * R1 * cos(θ_1)) + (M2 * R2 * cos(θ_2)) + (M3 * R3 * cos(θ_3)) + (M4 * R4 * cos(θ_4)) = 0

Given that M4 and θ_4 are unknowns and the center of mass must be at the center (x = 0), we can simplify to find R4:

M4 * R4 * cos(θ_4) = -((M1 * R1 * cos(θ_1)) + (M2 * R2 * cos(θ_2)) + (M3 * R3 * cos(θ_3)))

To determine R4 numerically, we need to choose an angle θ_4, and typically we would choose 0 degrees for simplicity since we are only looking for the horizontal component.

For simplicity, let's assume θ_4 = 0, which means R4 = -((M1 * R1 * cos(θ_1)) + (M2 * R2 * cos(θ_2)) + (M3 * R3 * cos(θ_3))) / M4

Plugging in the values, R4 = -((525 g * 13 cm * cos(35°)) + (325 g * 22 cm * cos(-45°)) + (225 g * 16 cm * cos(-205°))) / 875 g

After calculations using a calculator, we get R4 to be approximately -4 cm from the center of the tray. It is important to note that a negative position indicates that the glass must be placed on the opposite side of the tray relative to the positive x-direction.