You find that if you hang a 1.90 kg weight from a vertical spring, it stretches 2.90 cm .
Part A
What is the force constant of this spring in N/m?
Part B
How much mass should you hang from the spring so it will stretch by 9.06 cm from its original, unstretched length?
A. k = F/d = M*g/d = 1.9*9.8/0.029 = 642N/m.
B. 0.0906m * 642N/m = 58.2N.
Mass = 58.2/9.8 = 5.94kg.
To answer Part A, we need to use 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 written as:
F = -kx
Where:
F is the force exerted by the spring (in Newtons),
k is the force constant or spring constant (in N/m),
x is the displacement or change in length of the spring (in meters).
In this case, we know the mass of the weight hanging from the spring (1.90 kg) and the displacement of the spring (2.90 cm or 0.029 m).
First, let's convert the mass to force:
Weight = mass x gravity
Weight = 1.90 kg x 9.8 m/s^2 ≈ 18.62 N
Now, apply Hooke's Law to find the spring constant:
18.62 N = -k * 0.029 m
To solve for k, divide both sides of the equation by -0.029 m:
k = 18.62 N / -0.029 m ≈ -641.73 N/m
The force constant of this spring is approximately 641.73 N/m.
Moving on to Part B, we can use the same formula from Part A while solving for the mass.
We are given the new displacement or change in length of the spring (9.06 cm or 0.0906 m) and we need to find the mass required to achieve this displacement.
Again, apply Hooke's Law:
F = -kx
Now, solve for the mass:
mass x gravity = -kx
Divide both sides of the equation by gravity:
mass = (-kx) / gravity
Plug in the known values:
mass = (-641.73 N/m * 0.0906 m) / 9.8 m/s^2 ≈ -5.950 kg
The mass required to stretch the spring by 9.06 cm from its original, unstretched length is approximately 5.950 kg.