a spring of spring length 5times 10 tothe power of 4 N/m,is between a rigid beam and the output piston of a hydraulic lever.an mpty container with negligable lengthsits on th input piston.the input piston has an area of Ai and the output piston has area 22.0Ai,initially spring is at rest length.how many kg of sand must be slowly poured into the container to press thr spring by 7cm?
The length of the spring is not 5*10^4 N/m; that is its stiffness (spring constant), k. You need to be more careful stating the question. Please capitalize the first work in sentences and use spaces before new words.
The force needed to compress the spring X = 0.07 m will be
F = k*X = 0.07*5*10^4 = 3500 N
Take the hydraulic lever mechanical advantage of 22 into account when computing the required amount of sand. You will need a sand weight of only 3500/22 = 159.1 N to compress the spring the desired amount. Divide that by g for the mass of sand in kg.
To solve this problem, we need to understand the principles of Hooke's Law and Pascal's Law.
1. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. It can be expressed as F = k * x, where F is the force, k is the spring constant, and x is the displacement.
2. Pascal's Law states that the pressure exerted in a fluid is transmitted equally in all directions. This means that the pressure on the input piston of the hydraulic lever will be the same as the pressure on the output piston.
Now, let's calculate the force required to compress the spring by 7 cm.
1. Convert the displacement to meters:
x = 7 cm = 0.07 m
2. Use Hooke's Law to find the force:
F = k * x
F = (5 * 10^4 N/m) * (0.07 m)
F = 3500 N
Next, let's calculate the force exerted on the output piston using Pascal's Law.
1. The pressure on the input piston is the same as the pressure on the output piston, so:
P_input * A_input = P_output * A_output
2. The area of the output piston is 22.0 times the area of the input piston, so:
A_output = 22.0 * A_input
3. Substitute the values into the equation:
P_input * A_input = P_output * (22.0 * A_input)
4. Since the pressure is the force divided by the area, we can rewrite the equation as:
F_input / A_input = F_output / (22.0 * A_input)
Simplifying further:
F_input = F_output / 22.0
5. Substitute the force values:
F_input = 3500 N / 22.0
F_input = 159.1 N
Now, let's calculate the mass of sand required by considering the force exerted on the input piston.
1. The force exerted on the input piston is equal to the weight of the sand:
F_input = m * g
m = F_input / g
2. The acceleration due to gravity, g, is approximately 9.8 m/s^2.
3. Substitute the force value and calculate the mass:
m = 159.1 N / 9.8 m/s^2
m ≈ 16.23 kg
Therefore, approximately 16.23 kilograms of sand must be slowly poured into the container to compress the spring by 7 cm.