Why is the work done by tension on M1 half of the work done by gravity?

[Drakkith]

Homework Statement

A mass m1 = 5.1 kg rests on a frictionless table and connected by a massless string over a massless pulley to another mass m2 = 5 kg which hangs freely from the string. When released, the hanging mass falls a distance d = 0.83 m.

Homework Equations

The Attempt at a Solution

I don't need help with a specific answer, instead I need help understanding something.

The work done by gravity on the system is 40.71 J. (W = M₂gh, W = 5*9.81*0.83 = 40.71)
The work done by the tension of the string on M₁ is 20.57 J.
Why is the work done by tension on M₁ half of the work done by gravity? I only used the mass of M₂ to find the work done on the system by gravity, so shouldn't the work done by the tension be more since M₁ has more mass?

(On the other hand, gravity is the one pulling M₂ in the first place, which is supplying the tension on M₁. So confused...)

 

[Nathanael]

so shouldn't the work done by the tension be more since M₁ has more mass?

Be more than what? More than half? (It is more than half, not by much, but m₁ is not larger by much, either.)

 

[SammyS]

Homework Statement

A mass m1 = 5.1 kg rests on a frictionless table and connected by a massless string over a massless pulley to another mass m2 = 5 kg which hangs freely from the string. When released, the hanging mass falls a distance d = 0.83 m.

Homework Equations

The Attempt at a Solution

I don't need help with a specific answer, instead I need help understanding something.

The work done by gravity on the system is 40.71 J. (W = M₂gh, W = 5*9.81*0.83 = 40.71)
The work done by the tension of the string on M₁ is 20.57 J.
Why is the work done by tension on M₁ half of the work done by gravity? I only used the mass of M₂ to find the work done on the system by gravity, so shouldn't the work done by the tension be more since M₁ has more mass?

(On the other hand, gravity is the one pulling M₂ in the first place, which is supplying the tension on M₁. So confused...)

The tension also does work on m₂ . (This is a negative quantity.)

 

[Drakkith]

Be more than what? More than half? (It is more than half, not by much, but m₁ is not larger by much, either.)

More than the work done by gravity.
But of course that's impossible, as gravity is doing the work to begin with...

Am I correct in assuming that if M₂ wasn't bound to M₁, the work done by gravity would accelerate M₂ to a higher velocity than when M₁ and M₂ are connected, but the work done would still equal 40.71 J?

I'm sorry if this is confusing. It's a multi-step problem that I've gotten almost every single step wrong the first time through.

 

[SammyS]

The tension is an internal force as regards the entire system. Therefore, it does no net Work to the system as a whole.

 

[Drakkith]

Okay, so gravity is pulling down on M₂. M₂ wants to accelerate under the force of gravity, but the tension in the rope only allows it to accelerate if M₁ also accelerates. The force from M₂is transferred through the string to M₁ which accelerates under the force from tension. The acceleration of both blocks is the same and is equal to (M₂g)/(M₁+M₂), or 4.86 m/s².

The work done on each block is equal to their final kinetic energy and added together equals 40.71 J, which is the total work done on the system.

 

[Nathanael]

The tension does negative work on m₂ (it slows it down) and positive work on m₁ (speeds it up). You don't need to consider it to solve any problems because (as Sammy just said) it is zero for the entire system (it does the same amount of positive work on m₁ as it does negative work on m₂).

If (out of curiosity) you want to find the work done by tension (call it W_T) this is how I would find it:
(KE)₁=W_T
(KE)₂=W_g-W_T
(W_g is the work done by gravity and (KE)_n is the kinetic energy of block 1 and 2 respectively)

First note about this system of equations: If you look at the total energy, (KE)₁+(KE)₂, it is equal to the work done by gravity (tension does no work on the system).

Second note: if m₁=m₂ then (KE)₁=(KE)₂ (because they have the same speed) which leads to W_T=0.5W_g
(If the masses are the same the work done by tension is half the work done by gravity.)Since they have the same speed, the ratio of kinetic energies will be the ratio of their masses. Thus you have $\frac{m_2}{m_1}=\frac{W_g-W_T}{W_T}$

In this case, this equation leads to $W_T=\frac{W_g}{\frac{5}{5.1}+1}=\frac{W_g}{1.98}=$ slightly more than half the work done by gravity.
Again, none of this is important information. The fact that the work done by tension essentially makes it ignorable. I just wrote this to explain your observation that the work done by tension is (slightly more than) half.

 

[Nathanael]

Okay, so gravity is pulling down on M₂. M₂ wants to accelerate under the force of gravity, but the tension in the rope only allows it to accelerate if M₁ also accelerates. The force from M₂is transferred through the string to M₁ which accelerates under the force from tension. The acceleration of both blocks is the same and is equal to (M₂g)/(M₁+M₂), or 4.86 m/s².

The work done on each block is equal to their final kinetic energy and added together equals 40.71 J, which is the total work done on the system.

Yes I'd say this is a satisfactory understanding. Calling tension "the force from M₂" is a little iffy to me but you have the right idea.

(You said in your OP M₂ was "supplying the tension" which is also a little iffy to me.)

 

[Drakkith]

Yes I'd say this is a satisfactory understanding. Calling tension "the force from M₂" is a little iffy to me but you have the right idea.

(You said in your OP M₂ was "supplying the tension" which is also a little iffy to me.)

I'm sorry, I don't know how else to describe it. How would you?

 

[Nathanael]

I'm sorry, I don't know how else to describe it. How would you?

Yeah you have the right idea it doesn't really matter.

Instead of saying "The force from M₂ is transferred through the string to M₁" I would say something like "The tension is the same throughout the rope, so it is the same on M_1 and M_2"

Sorry I probably shouldn't have complained because I don't really have a good way to say it either

 

[SammyS]

Okay, so gravity is pulling down on M₂. M₂ wants to accelerate under the force of gravity, but the tension in the rope only allows it to accelerate if M₁ also accelerates. The force from M₂is transferred through the string to M₁ which accelerates under the force from tension. The acceleration of both blocks is the same and is equal to (M₂g)/(M₁+M₂), or 4.86 m/s².

The work done on each block is equal to their final kinetic energy and added together equals 40.71 J, which is the total work done on the system.

That looks very satisfactory to me.

 

[Drakkith]

Thanks guys!