What is the tension in the rope?

[Theorγ]

Homework Statement

A helicopter with mass 15,000 kg is lifting a truck with mass 4,500 kg up in the air with acceleration 1.4 m/s^2. The helicopter lifts using a rope. Calculate the tension of the rope and the net upward force of the helicopter.

m1 = mass of helicopter = 15,000 kg
m2 = mass of truck = 4,500 kg
a net = 1.4 m/s^2
g = -9.8 m/s^2

Homework Equations

F = ma
T = m(a + g)

The Attempt at a Solution

I first calculated normal and weight forces of the two objects, obtaining ±147,000 N for the helicopter and ±44,100 N. I then I tried to calculate the net upward force by subtracting the two opposing forces of the two objects:

Fn1 - Fg2 = 102900 N

The tension I tried to calculate by using the second equation:

T = m(a + g)
T = 4500 *( 1.4 + 9.8)
T = 50,400 N

So now can someone tell me if this is all correct? And also can someone properly explain to me how the tension equation is actually derived, because the main problem I encountered when solving this problem was that I didn't really know how tension was used, and resorted to cheating by using the equation above...

 

[Dougggggg]

Your answer for the tension looks good. As for how to derive that. If you take F=ma and look at all the forces acting on the truck, you have something like this.

T-mg=ma

Bring over the mg, factor out an m and you have T=m(a+g)

For the force on the helicopter, consider for a moment the F=ma formula for the helicopter. What are the forces (be sure to watch your signs)?

 

[Theorγ]

So the net upward force is the combination of the helicopter's normal force with normal force of the truck?

 

[SammyS]

So now can someone tell me if this is all correct? And also can someone properly explain to me how the tension equation is actually derived, because the main problem I encountered when solving this problem was that I didn't really know how tension was used, and resorted to cheating by using the equation above...

A Free Body Diagram for the truck would be a good place to start. Draw one. It should give the following.

The forces on the truck are:

A. The upward force which the rope exerts on the truck. This is equal to the tension, T.

B. The downward force which gravity exerts on the truck This is equal to m₂g.
​

Thus the net force, F_net on the truck is, F_net = T - m₂g.

Newton's 2^nd Law tells us that F_net = m₂a, the net force on the truck is equal to the mass of the truck, times the acceleration on the truck.

Thus we have:

T - m₂g = m₂a

Now, solve this for tension, T.

 

[Dougggggg]

Exactly what he said above ^^^^

When problems get more 2D and 3D, there is almost no way to do it without.

 

[Theorγ]

A Free Body Diagram for the truck would be a good place to start. Draw one. It should give the following.

The forces on the truck are:

A. The upward force which the rope exerts on the truck. This is equal to the tension, T.

B. The downward force which gravity exerts on the truck This is equal to m₂g.
​

Thus the net force, F_net on the truck is, F_net = T - m₂g.

Newton's 2^nd Law tells us that F_net = m₂a, the net force on the truck is equal to the mass of the truck, times the acceleration on the truck.

Thus we have:

T - m₂g = m₂a

Now, solve this for tension, T.

Oh I'm sorry for not mentioning, but the first guy who posted's response helped me understand the tension part, its just the net upward force that I'm stuck with now.

 

[Theorγ]

Can someone tell me if the following forces are correct for the helicopter:
A. The helicopter moves upward with the force m₁a
B. The gravitational force is downwards with m₁g
C. The tension is directed from downwards denoted by -F_T

Are theses all the forces affecting the helicopter?

 

[Dougggggg]

Ok now you are close.

A. The force I believe you are referring to is the lift force. If so, we are not certain if this is equal to m₁a. So instead you could just but that as some single variable.

B. Yes.

C. Yes, which according to Newton's 3rd law, is the T you had already solved for.

So now the sum of all of that is equal to m₁a.

 

[Theorγ]

So that means that the net upward force is equal to m₁a? Which leads to -F_T - m₁g = m₁a = 197400 N?

 

[NEILS BOHR]

yup
looks rite to me...

 

[Dougggggg]

So that means that the net upward force is equal to m₁a? Which leads to -F_T - m₁g = m₁a = 197400 N?

Your so close. Think about what your equation is saying though. It has a bunch of downward forces. Therefore it would have to be accelerating downward. We know that isn't the case. There is one upward force. The lift force of the helicopter. In fact, that is what you are trying to solve for. The net UPWARD force. You want to know what is the total force acting up. So you are solving for that lift force.

 

[Theorγ]

That means there must have been one more force that acted on the helicopters system? Leading to this equation:

-F_T - m₁g + F_Up= m₁a
F_Up= m₁a + F_T + m₁g
F_Up = 218400

Is this correct now?