河北卢龙：发展设施农业 推动乡村振兴(1)

Question

There are two people lifting a table of length $l$. One is short the other one is tall. The table is solid so its center of mass is therefore at the center of the table. Since the two people are of different heights, the table ends up being inclined $\theta$ with respect to the horizontal while lifted.

Is the shorter person exerting more effort?

sum of net forces: $\mathbf{F}_2 + \mathbf{F}_1 = mg$

sum of torques: $\mathbf{r}_2 \times \mathbf{F}_2 + \mathbf{r}_1 \times \mathbf{F}_1 = 0 \implies -\frac{l}{2}(\cos{\theta}, \sin{\theta})\times(0,F_2) + \frac{l}{2}(\cos{\theta}, \sin{\theta})\times(0,F_1) = \frac{l}{2}\sin\theta(-F_2+F1) = 0\implies F_2 = F_1$

The maths tell me that they are both doing exerting the same force. But when trying to replicate it in person it does feel heavier if i am the one doing $F_1$.

What am I missing?

Answer 1

If you both pull straight up, you're sharing the load exactly equally. As a matter of practical experience and biomechanics, though, you may end up bearing more weight on the lower end, since both parties may not be pulling straight up. Basically, the free-body diagram shown may not accurately depict how people actually carry things.

Imagine the scenario of carrying a sofa down a flight of stairs. The person at the bottom may carry more load, as they are likely bracing the couch with their body, pushing back up the stairs at an angle instead of vertically. The person at the top contributes a relatively smaller counter-rotating torque perpendicular to the couch, rather than gripping the corner of the upholstery and pulling straight up. It's possible the force diagram looks something more like this:

If you tilt the object even more, the person at the lower end can carry the entire weight of the object, with the person at the top end contributing arbitrarily little horizontal force to keep the object from toppling over.

The point of application of force will also have an effect for real objects with thickness, even if force is applied directly upwards. Consider what happens when two people carry a box down stairs, picking it up in the usual fashion by the bottom corners. As the box tilts, the lower person's corner gets closer to the center of mass and the upper person's corner gets farther - to balance the torque, the lower person must exert a greater force than the upper one:

This doesn't happen when considering an idealized flat object, but it certainly makes a difference when carrying real things.

Answer 2

Weight of a rigid object does not flow downhill. Gravity pulls down on each atom. The atoms all exert forces on each other to maintain their positions with respect to each other. Compare that to a bag of water. If you lift one end, the water flows away, and the other end gets heavier.

The floor holds up the table, exerting an upward force of half the table's weight on each end. If you lift one end, the whole table moves. You have to lift half the table's weight to get your end off the floor. The other half is still held by the floor. If you hold your end just off the floor, you are holding half the weight.

If you rotate the table, it gets easier. If you balance the table, the floor holds the entire weight.

But it doesn't have to be that way. You could pull upward on the balanced table and hold up half the weight. You could pull harder so the whole table is hanging from your hands. The floor will push hard enough to hold up any weight you don't hold up. The total upward force is equal to the weight of the table. It works this way only for a balanced table.

With two people, you might think the weight distribution isn't automatic like for the floor. All you can say is that if you are holding the table up, the total upward force is the weight of the table.

But if you exert different upward forces on opposite ends of the table, you create a torque. The table will rotate. It only stays at the same tilt if the upward forces are equal. See Toppling of a cylinder on a block for more.

For a balanced table, the two upward forces are in line. So the torque is $0$ even if the forces are different.

You do have to be careful to lift upward. If the tall person pulls toward himself he adds a horizontal component. His total force is larger than his upward force. And the short person must lift and pull horizontally toward himself as well as lift. Harder for him too.