Newton's 2nd Law: Why 10kg Mass Doesn't Fall Faster Than 5kg Mass

[Soaring Crane]

A 10 kg mass experiences twice the gravitational force as a 5 kg mass. Why doesn't the 10 kg mass fall faster than the 5 kg mass?


I never understood the reason. All I know is that Galileo found that objects dropped near the Earth's surface will fall faster with the same acceleration or g (neglecting air resistance).

Thanks.

 

[Pyrrhus]

Using Newton's 2nd Law, Neglecting Air Resistance

$$\sum_{i=1}^{n} \vec{F}_{i} = m \vec{a}$$

Forces acting on a body on freefall

$$m \vec{g} = m \vec{a}$$

so

$$\vec{g} = \vec{a}$$

All objects will experience the same acceleration, no matter what their mass is.

 

[Tide]

All I know is that Galileo found that objects dropped near the Earth's surface will fall faster with the same acceleration or g (neglecting air resistance).

Thanks.

Well, no, that discovery had to wait for Isaac Newton who recognized Earth's gravitational pull on the moon. Galileo was a little more down to Earth.

 

[KaneOris]

Either that is a trick question, or who ever made it up doesn't know very much

 

[Tide]

Either that is a trick question, or who ever made it up doesn't know very much

I don't think it's a trick. It is pedagogical and intended to get students to think it through.

 

[vsage]

A 10 kg mass experiences twice the gravitational force as a 5 kg mass. Why doesn't the 10 kg mass fall faster than the 5 kg mass?

I kind of like the way the question was worded. It leaves very little open to interpretation. I remember in my physics class last year the teacher asked a question like this but worded it as "two objects, one twice the mass of the other are dropped. how much longer does it each mass to reach the ground?" Of course the question didn't mention the shape of the objects, air resistance being negligible, etc. It's concise and none of that shape stuff applies with the above worded question because it's clear what the magnitude of the resultant force is.

Edit: Tide how'd you get those homework helper and science advisor pictures? :)

 

[Tide]

Vsage,

I agree with you on the wording!

Regarding the pics - they just appeared there one day. I guess I just post too much! ;-)

 

[Doc Al]

A 10 kg mass experiences twice the gravitational force as a 5 kg mass. Why doesn't the 10 kg mass fall faster than the 5 kg mass?

As others have explained, from Newton's 2nd law you know that the acceleration of an object is given by:
$$a = F/m$$
Since the gravitational force is proportional to the mass, that ratio never changes. Neglecting friction, all falling objects have the same acceleration due to gravity.

I never understood the reason. All I know is that Galileo found that objects dropped near the Earth's surface will fall faster with the same acceleration or g (neglecting air resistance).

Galileo found that all objects had the same acceleration, but he didn't know why. For that we had to wait for Newton.

In addition to his experimental work, Galileo made some clever arguments that showed that all objects must fall at the same rate. Here's the basic idea of one of them. Imagine two cannon balls falling side by side. They, of course, fall together. But if they were attached (thus making a double-sized ball) would it change anything? No! So two masses stuck together will fall at the same rate as one mass. Thus the acceleration must be independent of the mass.

 

[Doc Al]

Edit: Tide how'd you get those homework helper and science advisor pictures?

Those "ribbons" are awarded by the PF staff to recognize members who consistently prove themselves to be a reliable source of scientific information and who extend themselves to give good homework help. He earned it!

 

[tony873004]

A 10 kg mass experiences twice the gravitational force as a 5 kg mass. Why doesn't the 10 kg mass fall faster than the 5 kg mass?

I don't like the wording either.
The 10 kg mass will not experience twice the gravitational force, it will experience the same gravitational force as the 5 kg mass.

The combined gravitational force of the Earth + 10kg mass will be twice that of the combined gravitational force of the Earth + 5 kg mass. And that's why the 10kg mass will weigh twice as much as the 5kg miass when you place them on a bathroom scale.

But an object doesn't exert a gravitational force on itself. The force exerted
by the Earth will be the same on both objects.

 

[Doc Al]

The 10 kg mass will not experience twice the gravitational force, it will experience the same gravitational force as the 5 kg mass.

Not true. The gravitational force on the 10 kg mass is twice that on the 5 kg mass.

The combined gravitational force of the Earth + 10kg mass will be twice that of the combined gravitational force of the Earth + 5 kg mass. And that's why the 10kg mass will weigh twice as much as the 5kg miass when you place them on a bathroom scale.

I don't know what you mean by "combined gravitational force". Each mass and the Earth exert equal and opposite forces on each other. In the case of the 10 kg mass, that force is about 98 N; for the 5 kg mass, half that.

The force exerted
by the Earth will be the same on both objects.

Not true. (See above.)

 

[tony873004]

Not true. The gravitational force on the 10 kg mass is twice that on the 5 kg mass.

You're right. I was confusing it with acceleration.