A spring with a tension of 200N/mm is compressed for a distance of 150mm.
Determine the Work done if the spring is compressed a further 50mm.
200 N/mm = 200*10^3 Newtons/meter
so k = 2*10^5 Newtons/meter
find work to compress 0.150 meter
find work to compress 0.200 meter
difference is additional work to do last 0.050 meter
so
W = (1/2)kx^2
W = (1/2)(2*10^5)(0.20^2 - 0.15^2) Joules
work (stored energy) is equal to ... 1/2 * k * d^2
calculate work for 150 mm and 200 mm ... find the difference
To determine the work done when the spring is compressed a further 50mm, we can use the formula for elastic potential energy:
Elastic potential energy = 0.5 * k * x^2
Where:
k is the spring constant (tension) in N/mm,
x is the distance the spring is compressed in mm.
First, let's convert the values:
k = 200 N/mm = 200 N / 1000 mm (converting mm to meters)
x = 50 mm = 50 / 1000 m (converting mm to meters)
k = 0.2 N/m (converting N/mm to N/m)
x = 0.05 m (converting meters)
Now, let's calculate the work done:
Elastic potential energy = 0.5 * 0.2 N/m * (0.05 m)^2
Elastic potential energy = 0.5 * 0.2 N/m * 0.0025 m^2
Elastic potential energy ≈ 0.00025 J (Joules)
Therefore, the work done when the spring is compressed a further 50mm is approximately 0.00025 Joules.
The work done on a spring is equal to the change in the potential energy of the spring. The potential energy of a compressed or stretched spring can be calculated using the formula:
Potential energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement (change in length) of the spring.
In this case, the spring has a tension of 200 N/mm, which means its spring constant is 200 N/mm. However, we need to convert the displacement from millimeters to meters:
Displacement = 150 mm = 150/1000 m = 0.15 m
Using the formula for potential energy:
Initial potential energy = (1/2) * 200 * (0.15)^2
Now, the spring is compressed a further 50 mm, which means the new displacement is:
New displacement = 0.15 m + (50/1000) m = 0.2 m
To find the work done, we need to find the difference in potential energy before and after the additional compression. Thus, we need to calculate the new potential energy:
New potential energy = (1/2) * 200 * (0.2)^2
The work done is the difference in potential energy:
Work done = New potential energy - Initial potential energy
Plugging in the values:
Work done = [(1/2) * 200 * (0.2)^2] - [(1/2) * 200 * (0.15)^2]
Simplifying:
Work done = (1/2) * 200 * [(0.2)^2 - (0.15)^2]
Calculating:
Work done = (1/2) * 200 * [0.04 - 0.0225]
Work done = (1/2) * 200 * 0.0175
Work done = 1/2 * 200 * 0.0175
Work done = 1/2 * 3.5
Work done = 1.75 Joules
Therefore, the work done on the spring when it is compressed a further 50mm is 1.75 Joules.