A linear spring of stiffness k is designed to stop the 20-Mg railroad car traveling at 8 km/h within 400mm after impact. Find the smallest value of k that
will produce the desired result.
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To find the smallest value of k that will produce the desired result, we can use Hooke's Law, which states that the force exerted by a linear spring is directly proportional to the displacement from its equilibrium position.
1. Convert the mass of the railroad car from Mg (metric tons) to kg:
20 Mg = 20 * 1000 kg = 20,000 kg
2. Convert the velocity of the railroad car from km/h to m/s:
8 km/h = 8 * (1000 m/3600 s) = 2.22 m/s
3. Convert the displacement from mm to m:
400 mm = 400 / 1000 m = 0.4 m
4. Calculate the initial kinetic energy of the railroad car using the equation:
KE = (1/2) * mass * velocity^2
KE = (1/2) * 20,000 kg * (2.22 m/s)^2
5. Calculate the work done by the spring using the equation:
Work = force * displacement
Work = (1/2) * k * displacement^2
6. Equate the initial kinetic energy to the work done by the spring:
(1/2) * 20,000 kg * (2.22 m/s)^2 = (1/2) * k * (0.4 m)^2
7. Solve for k:
k = ((1/2) * 20,000 kg * (2.22 m/s)^2) / ((1/2) * (0.4 m)^2)
= (20,000 kg * 2.22^2 m^2/s^2) / (0.4^2 m^2)
= (20,000 * 4.9284) / 0.16
= 617,100 N/m
Therefore, the smallest value of k that will produce the desired result is 617,100 N/m.
To find the smallest value of k that will produce the desired result, we need to use the equation for the potential energy stored in a spring:
Potential Energy (PE) = (1/2) k x^2
Where:
PE is the potential energy stored in the spring,
k is the spring constant (stiffness),
x is the displacement of the spring.
In this case, the displacement (x) is given as 400mm, and the potential energy needs to be able to stop the 20-Mg (20,000 kg) railroad car traveling at 8 km/h.
First, we need to convert the displacement from millimeters to meters:
400mm = 400/1000 = 0.4m
Next, we need to convert the speed from kilometers per hour to meters per second:
8 km/h = 8 * 1000 / 3600 = 8/3.6 m/s
To stop the moving railroad car, the potential energy stored in the spring needs to be equal to the kinetic energy of the car before impact.
Kinetic Energy (KE) = (1/2) m v^2
Where:
KE is the kinetic energy of the car,
m is the mass of the car,
v is the velocity of the car.
Plugging in the values we have:
KE = (1/2) * 20,000 * (8/3.6)^2
Now, we set the potential energy of the spring equal to the kinetic energy of the car:
(1/2) k x^2 = (1/2) * 20,000 * (8/3.6)^2
Simplifying the equation:
k * x^2 = 20,000 * (8/3.6)^2
Now, we can solve for k:
k = 20,000 * (8/3.6)^2 / x^2
Plugging in the values:
k = 20,000 * (8/3.6)^2 / (0.4)^2
Calculating:
k = 20,000 * (2.22)^2 / (0.16)
Finally, we solve for k:
k = 20,000 * 4.92 / 0.16
k ≈ 615,000 N/m
Therefore, the smallest value of k that will produce the desired result is approximately 615,000 N/m.