A 64 kg woman stands on a very light, rigid board that rests on a bathroom scale at each end. She is 1.5m in on the 2m board. What is the reading on each of the scales in kg? Homework is almost due and I cant figure this out!!!!
To determine the reading on each of the scales, we need to first consider the forces acting on the board. The total weight of the woman, denoted as "F_woman", can be calculated as the product of her mass (64 kg) and the acceleration due to gravity (9.8 m/s^2).
F_woman = (64 kg) x (9.8 m/s^2) = 627.2 N
Next, we need to consider the torque or moment caused by the woman standing on the board. Torque is calculated by multiplying the force applied (F) by the distance from the pivot point (r). In this case, the pivot point is the point where the board rests on each scale.
The torque caused by the woman can be calculated as follows:
Torque = F_woman x r
Given that the woman is 1.5 m in from one end of the 2 m board, the distance from the pivot point can be calculated as:
r = 2 m * (1 - 1.5/2) = 1 m
Torque = (627.2 N) x (1 m) = 627.2 Nm
Since the board is in equilibrium (not rotating), the torques on both ends of the board must balance.
To find the reading on each of the scales, we can set up an equation using the torque.
Let S1 and S2 represent the readings on scale 1 and scale 2, respectively.
Torque_due_to_S1 = Torque_due_to_S2
Using the distance from the pivot point for each scale, we have:
S1 x 2m = S2 x 1m
Solving for S1 in terms of S2:
S1 = (S2 x 1m)/2m -- Equation (1)
We also know that the sum of the readings on the scales must equal the total weight of the woman (F_woman).
S1 + S2 = F_woman
Substituting Equation (1) into this equation:
(S2 x 1m)/2m + S2 = 627.2 N
Multiplying through by 2m to clear the denominator:
S2 + 2S2 = 1254.4 N
3S2 = 1254.4 N
S2 = 1254.4 N / 3
S2 ≈ 418.13 N
Substituting the value of S2 back into Equation (1):
S1 = (418.13 N x 1m) / 2m
S1 ≈ 209.06 N
Therefore, the reading on each scale is approximately 209.06 kg for scale 1 and 418.13 kg for scale 2.
To find the readings on each of the scales, we need to consider the forces acting on the system. The total weight of the woman and the board acts downwards at the center of mass of the system.
First, let's calculate the center of mass position of the woman and the board:
Center of mass position = (Mass of woman × distance from woman's end) + (Mass of board × distance from board's end) ÷ (Mass of woman + Mass of board)
Center of mass position = (64 kg × 1.5 m) + (0 kg × 0.5 m) ÷ (64 kg + 0 kg)
Center of mass position = 96 kg·m ÷ 64 kg
Center of mass position = 1.5 m
Since the center of mass is at 1.5 m from one end of the board, the forces exerted on the scales will depend on their distance from the center of mass.
Let's consider the forces acting on each scale:
Left Scale:
The woman's weight acts downwards from her end of the board. The board exerts an equal and opposite force on the woman towards the left. The bathroom scale at the left end exerts an upward force to balance the weight and the force exerted by the board.
Right Scale:
There is no weight or force acting from the board's end since it is very light and rigid. Therefore, the bathroom scale at the right end only has to balance the woman's weight.
Let's calculate the scale readings:
Left Scale:
The bathroom scale at the left end will read the normal force exerted by the board plus the normal force exerted by the woman. These two forces combined will balance the total weight acting downwards.
Normal force exerted by the woman = weight of the woman × g (acceleration due to gravity)
Normal force exerted by the board = weight of the board × g (acceleration due to gravity). Since the board is very light, it can be approximated as 0 kg.
Normal force exerted by the woman = 64 kg × 9.8 m/s^2
Normal force exerted by the board ≈ 0 kg × 9.8 m/s^2 ≈ 0 N
Therefore, the reading on the left scale will be the sum of the normal forces:
Reading on left scale = Normal force exerted by the woman + Normal force exerted by the board ≈ 64 kg × 9.8 m/s^2
Right Scale:
The bathroom scale at the right end will read the normal force exerted by the woman. This will balance the woman's weight.
Reading on right scale = Normal force exerted by the woman = weight of the woman × g (acceleration due to gravity)
Now you can substitute the values into the equations and calculate the readings on the scales.
Reading on left scale ≈ 64 kg × 9.8 m/s^2
Reading on right scale ≈ 64 kg × 9.8 m/s^2
Remember to use the appropriate units in your calculations. Once you've performed the calculations, you should have the readings on each scale in kilograms (kg).