At the bow of a ship on a stormy sea, a crewman conducts an experiment by standing on a bathroom scale. In calm waters, the scale reads 172 lb. During the storm, the crewman finds a maximum reading of 215 lb and a minimum reading of 131 lb. Calculate the maximum upward and downward acceleration experienced by the crewman.
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To calculate the maximum upward and downward acceleration experienced by the crewman, we can use the following formulas:
Maximum upward acceleration = (Maximum reading on the scale - Normal reading on the scale) / Normal reading on the scale
Maximum downward acceleration = (Normal reading on the scale - Minimum reading on the scale) / Normal reading on the scale
Using the given values:
Normal reading on the scale = 172 lb
Maximum reading on the scale = 215 lb
Minimum reading on the scale = 131 lb
Calculating the maximum upward acceleration:
Maximum upward acceleration = (215 - 172) / 172 = 0.25 or 25%
Calculating the maximum downward acceleration:
Maximum downward acceleration = (172 - 131) / 172 = 0.238 or 23.8%
Therefore, the maximum upward acceleration experienced by the crewman is 25% and the maximum downward acceleration is 23.8%.
To calculate the maximum upward and downward acceleration experienced by the crewman, we need to use Newton's second law of motion, which states that the force acting on an object is equal to its mass times its acceleration (F = ma). In this case, the force acting on the crewman is equal to his weight.
1. Firstly, let's convert the weights into forces by multiplying them by the acceleration due to gravity (g), which is approximately 9.8 m/s².
Weight in calm waters = 172 lb × 9.8 m/s² = 1685.6 N
Maximum weight during the storm = 215 lb × 9.8 m/s² = 2107 N
Minimum weight during the storm = 131 lb × 9.8 m/s² = 1283.8 N
2. Now, let's calculate the maximum upward acceleration and maximum downward acceleration.
Maximum upward acceleration:
The maximum upward force experienced by the crewman is the difference between the maximum weight during the storm and the weight in calm waters.
Maximum upward force = 2107 N - 1685.6 N = 421.4 N
Using Newton's second law (F = ma), we can rearrange the equation to find the maximum upward acceleration:
Maximum upward acceleration = Maximum upward force / mass
Since mass is not given and the same crewman is standing on the scale, we can assume that the mass does not change. Therefore, we can cancel it out from the equation.
Maximum upward acceleration ≈ Maximum upward force
Maximum upward acceleration ≈ 421.4 N
So, the maximum upward acceleration experienced by the crewman is approximately 421.4 N.
Maximum downward acceleration:
The maximum downward force experienced by the crewman is the difference between the weight in calm waters and the minimum weight during the storm.
Maximum downward force = 1685.6 N - 1283.8 N = 401.8 N
Again, using Newton's second law, we can find the maximum downward acceleration:
Maximum downward acceleration = Maximum downward force / mass
Following the same assumption as before, we can cancel out the mass from the equation.
Maximum downward acceleration ≈ Maximum downward force
Maximum downward acceleration ≈ 401.8 N
Thus, the maximum downward acceleration experienced by the crewman is approximately 401.8 N.
Please note that these calculations assume that the scale is ideal and accurately measures the force acting on the crewman in both calm and stormy conditions.
The man's mass is
M = 172/g = 5.34 slugs
The mass does not change with acceleration
Maximum upward acceleration "a" occurs when
M(a + g) = 215 lb
a + g = 215/172 = 40.3 ft/s^2
a = 40.3 - 32.2 = 8.1 ft/s^2
Maximum downward acceleration " a' " occurs when
M(g - a') = 131
g - a' = 24.5 ft/s^2
a' = 37.7 ft/s^2