When a mass of 20 gram is hung on a spring the length of the spring id 16 centimetres. Adding another 10 gram increases the length to 19 centimetres. What is the unstrecthed length of the spring
each 10 gram adds 3cm, so remove 30 grams, it goes from 19cm to 10cm, unstretched.
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To find the unstretched length of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, this can be expressed as:
F = -kx
Where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we have two different masses and their corresponding displacements:
Mass 1: m1 = 20 grams, displacement x1 = 16 centimeters
Mass 2: m2 = 30 grams (20 grams + 10 grams), displacement x2 = 19 centimeters
Let's calculate the force applied for each case using the equation above:
F1 = -kx1
F2 = -kx2
Now, we can set up a ratio of these forces to find the spring constant:
F1 / F2 = (m1 * g) / (m2 * g)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, we can cancel out the acceleration due to gravity, so the equation becomes:
F1 / F2 = m1 / m2
Simplifying further:
- kx1 / - kx2 = m1 / m2
x1 / x2 = m1 / m2
Now substitute the given values:
16 / 19 = 20 / 30
To find the unstretched length of the spring, we can set up another ratio using the displacement of the unstretched spring (x0) and the displacement with the mass m1 (x1):
x1 / x0 = m1 / m0
Where m0 is the mass of the unstretched spring. Since the displacement (x0) when there is no applied force is 0, the equation can be simplified to:
x1 / 0 = m1 / m0
To solve for m0, we can rearrange the equation:
m0 = (m1 * x0) / x1
Substituting the given values:
m0 = (20 grams * 0 centimeters) / 16 centimeters
Since the displacement of an unstretched spring is 0, we can conclude that the mass of the unstretched spring is 0 grams. Therefore, the unstretched length of the spring is independent of the mass hanging on it and is equal to 0 centimeters.