# Find the tension of a rope with a mass and spring attached

• valentina
In summary, in order to solve this problem correctly, you need to consider both the energy equation and the free body diagram, and take into account the acceleration of the mass.
valentina
Homework Statement
An object of mass m is being held by a spring without mass and with constant k, which at the same time is attached to a inextensible rope without mass, attached to the ceiling.

At first the spring is at its natural length, and it's released with no velocity.

Find the maximum value for the tension of the rope.
Relevant Equations
$$F=kx$$ (elastic force)

$$E_i=E_f$$
I'm having trouble with this problem, I think I solved it but I don't know if what I did is right...

At first when the velocity is 0 and the spring is at its natural length, there's just gravitational potential energy, so $$E_i=mgh$$

And then, when the mass is released and then reaches its minimum height (I choose $$h=0$$ for this case), it only has elastic potential energy and for a moment it doesn't have kinetic energy (right?), so that must be $$E_f=\frac{kh^2}{2}$$

By the the theorem of conservation of mechanical energy, we have $$E_i=E_f$$, then $$mgh=\frac{kh^2}{2}$$, so $$kh=2mg$$

Then I draw the free body diagram for both the mass and for the spring, here's where I don't know if what I did was right:

Where $$\vec{F}$$ is the elastic force

The equations I get are $$F=mg$$ and $$T=F$$ (because at that moment $$\vec{a}=0$$ ?)and this is where starts my confusion:

If I only use those two equations, I get that $$T=mg$$ BUT if I take into account the energy equation, I get that $$F=kh=2mg$$, and then $$T=2mg$$.

What would be the correct way to solve this problem? I'm really confused with the equations I got...

Delta2
valentina said:
because at that moment $$\vec{a}=0$$
Zero instantaneous velocity does not imply zero instantaneous acceleration. If a ball is thrown straight up and you consider the instant when the ball is at maximum height, what are the velocity and acceleration of the ball at that instant?

Leo Liu
valentina said:
What would be the correct way to solve this problem? I'm really confused with the equations I got...
The correct way to solve the problem is by ... (ahem) solving it correctly. Conservation of energy is one way. Drawing a free body diagram is another. If you draw the FBD, you need to understand that, for a spring-mass system, the acceleration is zero when the mass passes through the equilibrium position which is one amplitude away from the maximum extension (or compression) of the spring.

valentina said:
...
Then I draw the free body diagram for both the mass and for the spring, here's where I don't know if what I did was right:View attachment 275190View attachment 275191
Where $$\vec{F}$$ is the elastic force

The equations I get are ##F=mg## and ##T=F## (because at that moment $$\vec{a}=0$$ ?)and this is where starts my confusion:

If I only use those two equations, I get that $$T=mg$$ BUT if I take into account the energy equation, I get that $$F=kh=2mg$$, and then $$T=2mg$$.

What would be the correct way to solve this problem? I'm really confused with the equations I got...
I believe that your confusion started when assuming that ##F=mg## for height=0 in your diagram.
Rather, that force should be ##F=ma##.

The mass is experiencing two similar and mirroring conditions at both height=h and height=0: zero kinetic energy and maximum potential energy.

Maximum potential gravitational energy belongs to height=h.
Maximum potential elastic energy belongs to height=0.

There is a midpoint along that height at which all potential energy has become kinetic energy.
Because the mass has reached its highest velocity at that point, it can’t stop there.

The spring is still not acting on the mass at height=h; therefore, ##F=mg## is the only force inducing a downwards acceleration at that point.

The maximum value of the rope tension must be greater than mg for the mass to be stopped at height=0 and accelerated upwards again at same rate, in such way that a simple harmonic up-down motion is established.

http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1

Last edited:
Leo Liu and Delta2
valentina said:
The equations I get are $$F=mg$$ and $$T=F$$ (because at that moment $$\vec{a}=0$$ ?)and this is where starts my confusion:
##F\neq mg## because you didn't take the acceleration into consideration. The correct answer, in my opinion, is ##F-mg=ma##. Also, the acceleration of the mass is not 0, which was pointed out by Tsny.

etotheipi, Delta2 and Lnewqban

## 1. How does the mass of the object affect the tension of the rope?

The mass of the object attached to the rope will affect the tension of the rope. The heavier the object, the greater the tension in the rope will be. This is because the weight of the object creates a downward force that must be counteracted by the tension in the rope.

## 2. What role does the spring play in determining the tension of the rope?

The spring attached to the rope acts as a force that opposes the weight of the object. As the object pulls down on the rope, the spring will stretch and create an upward force that helps to balance out the tension in the rope. The stiffness of the spring will also affect the amount of tension in the rope.

## 3. How can I calculate the tension in the rope?

To calculate the tension in the rope, you will need to know the weight of the object and the stiffness of the spring. You can then use the formula T = mg + kx, where T is the tension, m is the mass of the object, g is the acceleration due to gravity, k is the spring constant, and x is the displacement of the spring.

## 4. Does the angle of the rope affect the tension?

Yes, the angle of the rope can affect the tension. The greater the angle of the rope, the greater the tension will be. This is because the weight of the object will be pulling more horizontally on the rope, resulting in a larger tension force needed to balance it out.

## 5. What factors can cause the tension in the rope to change?

The tension in the rope can change due to several factors, including the weight of the object, the stiffness of the spring, and the angle of the rope. Additionally, any external forces acting on the object or the rope, such as wind or friction, can also affect the tension.

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