To obtain the forces T and F exerted by the thumb and fingers, respectively, we can use the concept of torque equilibrium. The tray is being held parallel to the ground, which means it is not rotating. Therefore, the sum of the torques acting on the tray must be zero.
Let's calculate the torque due to the thumb first:
Torque due to the thumb (clockwise) = T * distance from the thumb to T
Since the tray is not rotating, the torque due to the food and cup must balance the torque due to the thumb:
Torque due to the food (counterclockwise) = (mass of food * g) * distance from the thumb to the plate
Torque due to the cup (counterclockwise) = (mass of cup * g) * distance from the thumb to the cup
Here, g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Now, let's set up the equation based on the torque equilibrium:
(T * 0.060) - [(1.00 * 9.8) * 0.240] - [(0.300 * 9.8) * 0.380] = 0
Simplifying the equation:
0.060T - 2.352 - 1.116 = 0
0.060T = 3.468
T ≈ 57.8 N upward (rounded to one decimal place)
Since the sum of the forces acting on the tray in the vertical direction is zero (the tray is not accelerating vertically), the force exerted by the fingers F can be found by subtracting the weight of the tray, food, and cup from the force exerted by the thumb.
Force F = (0.240 + 1.00 + 0.300) * g - T
F = (1.540) * 9.8 - 57.8
F ≈ 15.1 N upward (rounded to one decimal place)
Therefore, the force exerted by the thumb (T) is approximately 57.8 N downward, while the force exerted by the fingers (F) is approximately 15.1 N upward.