A 3 kg weight is placed upon a spring so that it hangs downward 12 centimeters. How far would the spring stretch if a 4 kg weight were placed upon it?
k = 3kg/12cm = 0.25 kg/cm.
d = 4kg * 1cm/0.25kg = 16 cm.
Or
d = (4kg/3kg) * 12cm = 16 cm.
To find out how far the spring would stretch if a 4 kg weight were placed upon it, we need to first understand the concept of Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Hooke's law can be mathematically expressed as:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant (a measure of the stiffness of the spring),
x is the displacement of the spring from its equilibrium position.
In this case, we can assume that the spring follows Hooke's law, which means that the displacement of the spring is directly proportional to the force applied.
Given the following information:
Mass of the weight (m1) = 3 kg
Displacement of the spring (x1) = 12 cm
Mass of the second weight (m2) = 4 kg
We can assume that the force applied by the first weight is equal to the force applied by the second weight, as both forces are balanced by the spring.
Since force = mass × acceleration, the force applied by the first weight, F1 = m1 × g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
Similarly, the force applied by the second weight, F2 = m2 × g.
Since the forces in both cases are the same, we can set up the following equation:
F1 = F2
m1 × g = m2 × g
Substituting the given values, we have:
3 kg × 9.8 m/s² = 4 kg × 9.8 m/s²
This equation tells us that the forces applied by both weights are equal.
Now, since the displacement of the spring is directly proportional to the force applied, we can set up the following proportion:
x1 / F1 = x2 / F2
Substituting the values we know:
12 cm / (3 kg × 9.8 m/s²) = x2 / (4 kg × 9.8 m/s²)
Simplifying, we can convert centimeters to meters:
0.12 m / (3 kg × 9.8 m/s²) = x2 / (4 kg × 9.8 m/s²)
Now, we can solve for x2, the displacement of the spring when a 4 kg weight is placed upon it:
x2 = (0.12 m / (3 kg × 9.8 m/s²)) * (4 kg × 9.8 m/s²)