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!!!!
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To find the reading on each of the scales, we can start by considering the forces acting on the woman and the board.
1. First, let's determine the weight of the woman. We are given that the woman's mass is 64 kg. The weight of an object can be calculated by multiplying its mass by the acceleration due to gravity, which is approximately 9.8 m/s^2. Therefore, the weight of the woman is:
Weight = mass * acceleration due to gravity
= 64 kg * 9.8 m/s^2
= 627.2 N
2. Next, let's consider the forces acting on the board. Since the board is at rest, the sum of the forces acting on it should be zero (according to Newton's first law of motion). There are three forces acting on the board: the woman's weight, the weight supported by the bathroom scale at one end, and the weight supported by the bathroom scale at the other end.
3. The total weight supported by the board is equal to the sum of the woman's weight and the net force exerted on the board by the bathroom scales. Since the net force is zero (according to the condition of the board being at rest), the total weight supported by the board is equal to the weight of the woman.
4. However, the distribution of this total weight is not uniform along the board. The weight distribution is based on the position of the woman on the board. In this case, the woman is 1.5 m in on the 2 m board, meaning that she is closer to one end.
5. To determine the readings on each scale, which represent the force exerted by each scale, we need to consider the weight distribution. Given that the board is 2 m long, the distance of the woman from the end of the board is 2 m - 1.5 m = 0.5 m. Thus, the distance of the woman from the other end of the board is also 0.5 m.
6. Since the board is at rest, the forces exerted by each scale should be equal and opposite in direction to the individual weights distributed along the board. Let's denote the reading on one scale as F1 and the reading on the other scale as F2.
7. Using the principle of moments (based on the concept of torque), we can calculate the readings on each scale. The moment of an object can be calculated by multiplying the force on the object by its distance from a pivot point.
8. Therefore, the moment created by the woman's weight (627.2 N) is equal to the moment created by the combined readings on the two scales. The moment created by the woman's weight is determined by multiplying the weight by the distance from the pivot (0.5 m), while the moment created by the scales can be calculated by multiplying F1 by its distance from the pivot (2 m - 1.5 m = 0.5 m) and F2 by its distance from the pivot (1.5 m).
9. Formulating this equation, we have:
Moment created by woman's weight = Moment created by scale 1 + Moment created by scale 2
(627.2 N) * (0.5 m) = F1 * (0.5 m) + F2 * (1.5 m)
10. Simplifying this equation, we get:
313.6 N*m = 0.5F1 N*m + 1.5F2 N*m
11. Since the scales measure force (in newtons), we can convert the moments to forces by dividing by the corresponding distances. Dividing both sides of the equation by 0.5 m, we have:
627.2 N = F1 + 3F2
12. Now we have two equations (based on the force and moment balances) to solve for F1 and F2:
F1 + F2 = 627.2 N
F1 + 3F2 = 627.2 N
13. Solve these equations simultaneously to find the readings on each of the scales. Subtracting the first equation from the second, we get:
2F2 = 0
This implies that F2 = 0. Therefore, substituting this value into the first equation, we have:
F1 + 0 = 627.2 N
F1 = 627.2 N
14. The reading on scale 1 (F1) is 627.2 N, or approximately 63.9 kg (since 1 N ≈ 0.102 kg).
15. Since F2 = 0, scale 2 does not exert any force. Therefore, the reading on scale 2 is 0 kg.
In summary, the reading on scale 1 is approximately 63.9 kg, and the reading on scale 2 is 0 kg.
To solve this problem, we need to consider the forces acting on the board.
Let's start by finding the total weight of the woman. We have given her mass as 64 kg, and since weight is the force due to gravity acting on an object, we can calculate it using the formula:
Weight = mass × gravitational acceleration
The standard gravitational acceleration is approximately 9.8 m/s². Therefore, the woman's weight is:
Weight = 64 kg × 9.8 m/s² = 627.2 N
Now, let's consider the forces acting on the board. Since the board is at rest, the forces on both ends of the board must be equal in magnitude and opposite in direction to balance it.
Let's assume the scale on the left end of the board is S1, and the scale on the right end is S2. Since the woman is standing 1.5m in on the 2m board, we can calculate the distances between the left and right end and the woman's position:
Distance from the left end = 2m - 1.5m = 0.5m
Distance from the right end = 1.5m
To calculate the reading on each scale, we need to distribute the woman's weight according to the distances. Assuming the left side of the board is positive and the right side is negative:
S1 × distance from the left end + S2 × distance from the right end = Total weight
S1 × 0.5m + S2 × (-1.5m) = 627.2 N
Now, let's solve for S1 and S2:
0.5S1 - 1.5S2 = 627.2 N
Since we have two unknowns, we'll need one more equation to solve the system. We can consider the sum of the forces acting on the board to be zero:
S1 + S2 = Total weight
S1 + S2 = 627.2 N
To find S1 and S2, we can use the simultaneous equation solver. Plugging in the values, we get:
S1 = 627.2 N - S2
(627.2 N - S2) + S2 = 627.2 N
627.2 N = 627.2 N
Therefore, both scales have readings of 627.2 N or approximately 64 kg each.
Remember that the reading on a scale is equivalent to the force acting on it, so the reading on each scale is 64 kg.