- Good morning.
Bobby, could you please read the problem and Billy could you please translate?
Flipping Physics
- A vertically hanging spring with a natural length of
5.4 centimeters is extended to a length of 11.4 centimeters when 25 grams is suspended from it.
What is the spring constant of the spring?
- The initial length of the spring is 5.4. centimeters,
the final length of the spring is 11.4 centimeters, the hanging mass is 25 grams.
We should convert those known values to base SI units
by multiplying the lengths by 1 meter divided by 100 centimeters
and the mass by 1 kilogram over 1,000 grams.
We are solving for the spring constant so lowercase k equals question mark.
- Thank you. Bo, please begin solving the problem.
- Um, sure...
- I bet this has to do with forces.
I think you should draw a free body diagram of the forces acting on the hanging mass.
- OK, force of gravity is down, force applied is down, force normal is up, and the force of the spring is up.
- Okay, let's talk about this free body diagram.
The force of gravity is caused by the pull of the earth on the hanging mass.
Bo, what causes the force applied?
- The mass causes the force applied.
- Actually, no, as I just explained, the mass causes the force of gravity.
- Right, so there is no force applied.
What surface is causing the force normal?
I guess there is no surface causing a force normal.
So the only forces are the force of gravity down and the spring force up, sorry!
- It's okay, we have the correct free body diagram now; however,
I want to make sure we fully understand the direction of the spring force.
The equation for a spring force is....
the force of the spring equals
the negative of the spring constant times the displacement from equilibrium position.
Remember, force and displacement are both vectors.
Billy, what does the negative in the equation mean?
- The negative means the direction of the force of the spring is opposite
the direction of the displacement of the spring.
Oh, that is how we know the direction of the spring force in the free body diagram.
The displacement of the spring is down, so the force of the spring needs to be in the opposite direction or up.
- Correct. The displacement of the spring is down and
according to the negative in the spring force equation, the direction of the force of a spring is
opposite the direction of the displacement of the spring.
So the spring force is up.
Bobby, please continue solving the problem.
- Now we can use Newton's second law in the y-direction.
The net force in the y-direction equals the spring force, which is positive because it's up,
minus the force of gravity, minus, because the force gravity is down,
the net force in the y-direction also equals mass times acceleration in the y-direction.
The hanging mass is now at rest, so the acceleration in the y-direction equals zero.
We can add the force of gravity to both sides so the spring force equals the force of gravity,
and we can now substitute in equations.
The force of the spring equals negative k times x
or the negative of the spring constant times the displacement from rest position.
And the force of gravity equals mass times acceleration due to gravity.
- Okay, please pay attention because a lot of students make this mistake.
When we drew the freebody diagram and summed the forces,
we already determined the direction of the spring force.
Again, we used the negative in the spring force equation
to determine the direction of the spring force in the free body diagram.
So we do not need to use the negative again.
Instead, when plugging the spring force equation into Newton's second law,
we need to use the magnitude of the spring force equation.
In other words, we use just k times x and not negative k times x.
Billy, please finish the problem.
- Now, we can divide both sides by the displacement from equilibrium position and substitute in numbers.
Mass is 0.025 kilograms, acceleration due to gravity is 9.81 meters per second squared,
the displacement from the equilibrium position..
Oh, we need to solve for that.
That would be the final length minus the initial length, or 0.114 minus 0.054 which is 0.060 meters.
And that gives us a spring constant of 4.0875 or 4.1 Newtons per meter with two significant digits.
- Thank you very much for learning....
- Mr. P?
- Yes, Bo?
The mass is displaced downward,
so the displacement from rest position should be negative 0.060 meters, right?
- Uh, OK.
Again, we already determined to the direction of the spring force in the free body diagram,
so we do not plug in a negative again.
Instead, we use the magnitude of the displacement from equilibrium position in the spring force equation.
So we use positive 0.060 meters even though the displacement is negative 0.060 m.
We do not want to reverse the direction by plugging in a negative.
Thank you very much for learning with me today.
I enjoyed learning with you.