# Two people are holding either end of a couch, is one person exerting more force than the other?

I was carrying a couch with my flatmate yesterday, and I started thinking about this.

Often when carrying objects like this, one person will be taller and has thus lifted his end of the couch higher than the other person's. Additionally, one can intentionally lift his end of the couch higher, for comfort.

My question is, in situation is one person carrying more weight than the other?

• That sounds like a nice homework question, doesn't it? What did you do to solve it? – CuriousOne May 31 '15 at 1:03
• @CuriousOne It's not a homework question. I was carrying a couch and starting wondering if my lifting it higher was making it harder for my flatmate. – dwjohnston May 31 '15 at 7:22

I believe the following diagram tells you everything you need to know: when the center of mass is above the support point, then the lower point will carry more of the weight since it is acting closer to the center of mass ($x_1 < x_2$) and torque balance requires that $F_1 x_1 = F_2 x_2$. Conversely, if the center of mass is below the support, the higher point will carry more of the weight when the object is tilted.

Understanding this diagram really helps when you are moving large objects up or down the stairs: the lower person should grab the object higher up (say the arms of the couch) if you want to achieve more equal weight distribution.

In the second case, if the person on the right pushes against the couch, it is possible for the person on the left to end up carrying the entire weight of the couch. In the case of carrying things up stairs that can easily happen... Which is why movers often use webbing straps to permit/ensure a more equal distribution of effort (and to allow lifting with shoulders/back/knees and not just arms).

• Nice answer, but this assumes that both people only apply vertical force. In reality, this is only true for 'level' carrying. Once you go up the stairs, the 'top' person will usually apply a force approximately perpendicular to the couch to keep it pointing in the right direction, and will not 'tug' at the couch since that would be very inefficient. Then, the picture will also make sense for slender beams, where you will also notice that the bottom person carries most of the weight (extreme case: near-vertical lifting) – Sanchises May 31 '15 at 14:53
• @sanchises the question explicitly stated "Is one person carrying more weight" which is (intentionally or not) a question strictly about vertical force (weight). I agree that real lifting often contains non-vertical components. – Floris May 31 '15 at 14:56
• I do not think it is intentional - it springs from a real-world example, and 'carrying weight' is non-physics English for 'Applying a force with a relatively strong vertical component'. Either way, I'll make my own drawing and we'll let the OP decide. – Sanchises May 31 '15 at 15:00
• @sanchises I have added an update see what you think. – Floris May 31 '15 at 15:05

Although Floris made a clear picture involving only vertical forces, this picture is mostly useful when carrying washing machines or large chairs, where the 'height' of the object is more pronounced.

However, you will find that, even when the object is mostly flat, the bottom person will carry most of the weight. The key here is the direction of the forces involved.

You will find that it's very difficult to lift a couch underneath you, 'tugging' it upwards. Instead, the top person will apply force in the direction that does not require any tugging (especially when you can't get a grip): perpendicular to the object. All you do, is offset the torque gravity incurs on the couch, where we take the bottom person as the pivot point. In the image above, the torque is provided by the component of gravity perpendicular to the couch (green arrow).

Since the top person is twice as far away from the pivot point as the center of gravity, he will only need to apply half of that component which was already less than the total gravity - i.e., much less than half of the total gravitational force. The top person has the easy job, although usually in a very awkward position.

The bottom person needs to account for the rest of the upward force required, along with a small horizontal component to make sure there is no horizontal acceleration (blue arrows) - i.e., a lot more, which is why being the bottom person is the heavier job.

• You make a valid point about the extreme case of a staircase, and the fact that in this position the push by the higher person will make the lower person's task harder - even for flat objects. Note that the original question was about the case of a couch being carried by people of unequal height - so the "stairs" argument didn't really play there. Giving you an upvote all the same... – Floris May 31 '15 at 15:37

I suspect the person lifting their side higher usually supports less weight, but it can go either way depending on how they push/pull.

When the couch is level, suppose both people support the same weight. Because the couch isn't rotating, the people are supporting it at points equidistant from the couch's center of mass, ensuring they exert no net torque.

Next we tilt the couch. The people are still supporting it equidistant from the center of mass. If the forces were all vertical, they would both need to lift the same weight as before because the net torque still needs to be zero.

However, it is likely in this case that the people exert equal and opposite horizontal forces as well. For the flat couch this doesn't matter; the horizontal forces are aligned and exert no net torque. For a tilted couch, if we suppose the horizontal forces are toward the center of mass of the couch (IE the person in front pushes backwards and vice versa), this would tend to torque the couch up to an even higher angle.

Since the couch isn't rotating, there must be some counter-torque. This comes from the lower side increasing its vertical force and the upper side decreasing it. So in this case the lower side supports more weight.

If the people are pulling on the couch (IE the person in front exerts net forward force) it's the opposite, and the person with the higher end supports more weight.

Note: this differs somewhat from Floris' answer due to different assumptions about the geometry; I agree with Floris' answer as well.

• This answer says the same thing as the upvoted answers; why is it downvoted? – Mark Eichenlaub Jun 1 '15 at 14:33

If you move heavy appliances or furniture, you'll realize that simple free body diagrams do not capture all the factors. Just imagine 20 feet long counch, rotated upward a log, e.g., 60 degrees. Of course the person at the bottom will bear most of the weight. The free body diagram only applies if couch is suspended using hinged anchor points at either end of the couch. Even in that contrived situation, if there is any nontrival clearance between the bearing and the rod at the 2 hinges, one hinge will bear more weight unless the distance between the centres of the two rods is exactly the same as the distance between the centres of the bearings.

• -1 A simple free body diagram can easily capture the forces involved in lifting a couch. It's all about the direction and position of the forces, but that is exactly the reason why people use FBD's at all! – Sanchises May 31 '15 at 14:42
• FBDs work well in space, or in contrived textboook problems. The model is easy, but it doesn't represent this particular reality. Did you read my example as to why? Do you actually move heavy furniture up stairs? I am concerned that this voting down without justification or, more importantly, trying to recconcile simple models with real world experience will lead to high ranked answers that are completely divorced from reality. I noticed that Mark E also got voted down simply because his honest attempt to impart reality to the problem complicates the misleadingly simple model. – user36800 May 31 '15 at 15:09
• I have plenty of experience moving, thank you. A free body diagram is a very powerful tool if you use it correctly, and is perfectly sufficient to depict all forces involved in moving a couch, as long as you do not make the wrong assumptions. I have dealt with a lot more complex problems that could still be depicted in a FBD during my masters in engineering. Don't say a modeling tool is bad just because you do not know how to use it properly. – Sanchises May 31 '15 at 15:31
• I see you added a higher fidelity model to address the deficiencies of the original answer. You're welcome. – user36800 May 31 '15 at 15:35
• From your experience, then you should know if you're helping to move a couch up some stairs, your contribution is far from just standing there for the couch to lean against you. I also find it disturbing that both people who tried to bring attention to a fundamental problem in the initial answer got downvoted without justification. In this particular case, you decided to focus on the opening sentence, misquoted it as a disparaging statement, and apparently didn't read it the actual content. It's obviously better just to leave problematic answers as-is. – user36800 May 31 '15 at 17:03