A particle of mass 0.2kg is found to stretch a certain spring by 5.0 cm.
a. What is the force constant of the spring?
b. By how much will a 1kg mass stretch the spring.
k=mg/.05 N/m
xnew=newForce/k
To determine the force constant of the spring and calculate the stretch caused by a 1kg mass, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement it undergoes. The formula for Hooke's Law is:
F = -kx
where:
F is the force exerted by the spring,
k is the force constant of the spring,
x is the displacement of the spring from its equilibrium position.
Let's solve each part of the question step by step:
a. What is the force constant of the spring?
Step 1: Convert the displacement from centimeters to meters.
Given that the displacement of the spring is 5.0 cm, we need to convert it to meters by dividing it by 100:
x = 5.0 cm / 100 = 0.05 m
Step 2: Calculate the force constant.
Using Hooke's Law (F = -kx), we can rearrange the formula to solve for the force constant (k):
k = -F / x
Since the mass of the particle is not given, we need to calculate the force exerted by the spring using Newton's second law of motion:
F = m * g
where m is the mass of the particle and g is the acceleration due to gravity (approximately 9.8 m/s²).
Assuming the particle is in equilibrium (no net force acting on it), the force exerted by the spring is equal to the weight of the particle.
F = m * g
Given that the mass of the particle is 0.2 kg, we can calculate the force:
F = 0.2 kg * 9.8 m/s² = 1.96 N
Now we can substitute the values into the formula to calculate the force constant (k):
k = -1.96 N / 0.05 m
k = -39.2 N/m
Therefore, the force constant of the spring is 39.2 N/m.
b. By how much will a 1kg mass stretch the spring?
Using Hooke's Law (F = -kx), we need to solve for x, the displacement of the spring caused by a 1kg mass.
Step 1: Calculate the force exerted by the 1kg mass.
Using Newton's second law of motion (F = m * g), we can find the force:
F = 1 kg * 9.8 m/s² = 9.8 N
Step 2: Rearrange Hooke's Law to solve for x:
F = -kx
x = -F / k
Substituting the values into the formula:
x = -9.8 N / 39.2 N/m
x = -0.25 m
Note that the negative sign indicates that the spring is being compressed rather than stretched.
Therefore, a 1kg mass will compress the spring by 0.25 meters.