Does a body behave as a point mass even at rest?

In summary: If the body is non-rigid, you will also need to ditch the idea of a single rotation rate multiplied by a fixed moment of inertia and stick with the sum of......the external forces acting on the body.Yes. That's generally good advice.
  • #36
mark2142 said:
I am trying an experiment but I don't know how can I apply constant force each time.
Possibly it is time to actually learn the math. It is not like ##\sum F_\text{ext} = m_\text{tot} a_\text{cm}## is terribly difficult.

Then you can do a thought experiment. Consider a bar-bell type object. A thin and nearly massless rod connecting two massive balls that are small compared to the length of the rod.

Apply (on paper) a small force on the right-hand ball for a small time.
Consider the resulting motion of the assembly. How fast does the center of mass move? In what direction does the center of mass move?
 
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  • #37
One should also be careful, which external force you consider. All you discuss here is about the motion in the homogeneous gravitational field close to Earth.

To that end consider the body under consideration (it can be an elastic or rigid solid body or even a fluid) as a discrete set of point particles, held together by interaction forces, which can be described as "pair forces", i.e., on the particle with the label ##j## the total force is given by
$$\vec{F}_j=\sum_{k \neq j} \vec{F}_{jk} + m_j \vec{g},$$
where ##\vec{g}## is the constant gravitational acceleration on due to the Earth (an approximation valid for motions close to Earth). The equations of motion for the particles thus read
$$m_j \ddot{\vec{x}}_j = \sum_{k \neq j} \vec{F}_{jk} + m_j \vec{g}.$$
Now we sum this equation over ##j## and use Newton's Third Law ("lex tertia"), according to which ##\vec{F}_{jk}=-\vec{F}_{kj}##, i.e., if the interaction force on particle ##j## due to the presence of particle ##k## is ##\vec{F}_{jk}## then the interaction force on particle ##k## due to the presence of particle ##j## is opposite. This means that by summing over ##j## all the interaction forces cancel and you get
$$\sum_j m_j \ddot{\vec{x}}_j=M \ddot{\vec{x}}_{\text{cm}} =M \vec{g},$$
where
$$M=\sum_j m_j, \quad \vec{x}_{\text{cm}}=\frac{1}{M} \sum_j m_j \vec{x}_j.$$
Our calculation shows that indeed the center of mass moves like a point particle in the gravitational field of the Earth,
$$\ddot{\vec{x}}_{\text{cm}}=\vec{g}=\text{const}.$$
For all other forces, and even the gravitational interaction with the Earth when considering the position dependence, is not as simple, because then you have (treating the Earth as a "point particle" sitting fixed at the origin of the coordinate frame),
$$m \ddot{\vec{x}}_j = \sum_{k \neq j} \vec{F}_{jk} -\gamma m_j m_{\text{Earth}} \frac{\vec{x}_j}{r_j^3} \quad \text{with} \quad r_j=|\vec{x}_j|.$$
Then summing over ##j## gives
$$M \ddot{\vec{x}}_{\text{cm}} = -\gamma m_{\text{Earth}} \sum_j \frac{m_j \vec{x}_j}{r_j^3}=\vec{F}_{\text{ext}},$$
and ##\vec{F}_{\text{ext}}## cannot be expressed in terms of ##\vec{x}_{\text{cm}}##!
 
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  • #38
jbriggs444 said:
If you look at the pencil tip where the force is applied, you can see that because the pencil both translates and rotates that the pencil tip moves farther than it would under either effect alone. So the work done and the kinetic energy gained will be greater than it would be under either effect alone. A force with the same magnitude can have greater effect because it acts over a greater distance.
jbriggs444 said:
We need more energy. We do not need more force.
jbriggs444 said:
But if we apply the force to the end of the pencil, the same force will be applied to a point on the pencil that moves for a greater distance. That means that the same magnitude of force can do a greater amount of work. It also means that we will have to put more effort into pushing the pencil on its end to maintain the same magnitude of force.
jbriggs444 said:
Intuitively, it is easier to push hard on a bowling ball than it is to push hard on a golf ball. The golf ball moves away fast when you push on it hard. It is difficult to keep up.
jbriggs444 said:
Both bowling ball and golf ball get the same momentum. But the golf ball gets 100 times more energy.
I think I got you. Its because its difficult to keep up at the end of a pencil(less mass) than at COM(more mass) when applying the force on its end that we are unable to provide it same momentum. We are unable to apply force for the required time. That's is why my glass cone on water experiment is failing. It is not covering the same distance when applying force off center.
Similarly the work done also becomes less since force is applied for lesser distance and so reduced KE for rotating object. Difficult to do. Yes?
 

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  • #39
It takes time to connect the dots. Thank you guys @hutchphd @jbriggs444 and everyone else. That was some new and interesting information not given in a book.
 
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<h2>1. What is a point mass?</h2><p>A point mass is a theoretical concept in physics that represents an object with no physical size or dimensions. It is often used to simplify calculations and models for objects with negligible size or mass compared to other objects in the system.</p><h2>2. How does a body behave as a point mass?</h2><p>A body behaves as a point mass when its size and shape can be ignored and its mass is concentrated at a single point. This means that the body's motion and interactions with other objects can be described using only its mass, position, and velocity.</p><h2>3. Does a body always behave as a point mass?</h2><p>No, a body does not always behave as a point mass. This concept is most applicable to objects with negligible size or mass, such as a small particle or a planet compared to the size of the universe. For larger objects, the effects of their size and shape cannot be ignored and must be taken into account in calculations.</p><h2>4. Can a body behave as a point mass even at rest?</h2><p>Yes, a body can behave as a point mass even when it is at rest. As long as the body's size and shape can be ignored and its mass is concentrated at a single point, its behavior can be described using the concept of a point mass. This is often the case for objects at rest on a surface, such as a book on a table.</p><h2>5. Are there any limitations to the concept of a point mass?</h2><p>Yes, there are limitations to the concept of a point mass. It is an idealized model that does not take into account the effects of an object's size and shape. In reality, all objects have some physical size and cannot be treated as a single point. Additionally, the concept of a point mass does not consider the internal structure of an object, which can affect its behavior in certain situations.</p>

1. What is a point mass?

A point mass is a theoretical concept in physics that represents an object with no physical size or dimensions. It is often used to simplify calculations and models for objects with negligible size or mass compared to other objects in the system.

2. How does a body behave as a point mass?

A body behaves as a point mass when its size and shape can be ignored and its mass is concentrated at a single point. This means that the body's motion and interactions with other objects can be described using only its mass, position, and velocity.

3. Does a body always behave as a point mass?

No, a body does not always behave as a point mass. This concept is most applicable to objects with negligible size or mass, such as a small particle or a planet compared to the size of the universe. For larger objects, the effects of their size and shape cannot be ignored and must be taken into account in calculations.

4. Can a body behave as a point mass even at rest?

Yes, a body can behave as a point mass even when it is at rest. As long as the body's size and shape can be ignored and its mass is concentrated at a single point, its behavior can be described using the concept of a point mass. This is often the case for objects at rest on a surface, such as a book on a table.

5. Are there any limitations to the concept of a point mass?

Yes, there are limitations to the concept of a point mass. It is an idealized model that does not take into account the effects of an object's size and shape. In reality, all objects have some physical size and cannot be treated as a single point. Additionally, the concept of a point mass does not consider the internal structure of an object, which can affect its behavior in certain situations.

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