F=ma for extended objects and movement of CM

In summary, the sum of forces acting on a particle, as determined by Newton's 2nd law, will determine its linear acceleration. For extended objects, such as a solid uniform rod, the sum of forces will determine the linear acceleration of its center of mass. This is also true for an arbitrary point on the rod. Torque may also play a role in determining the angular acceleration of the rod. The orientation of the forces may affect the resulting motion, with perpendicular forces causing rotational acceleration and inertial forces causing the rod to behave like a pendulum.
  • #1
mfactor
6
0
OK, so for a small particle, [itex]\Sigma F=ma [/itex]is very straight forward. If two forces are applied to that particle, the particle accelerates, which is determined by the summation of the forces applied.


But what if the object is extended one, say, a solid uniform rod (1 kg). Also, imagine that I applied 6N on the left end and 5N on the right end, perpendicular to the rod, in opposite directions. (see attached picture).

What would be the resulting linear acceleration of the center of mass? Shouldn't F=ma still be applicable for center of mass only?
(Obviously torque is involved in this case also, but should it affect the linear accleration of the CM?)

how about an arbitrary portion of the rod (not center of mass)?


Picture: The red arrow up is 6N. The blue arrow down is 5N. The grey stick is the solid rod.
 

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  • #2
The sum of the forces will determine, via Newton's 2nd law, the linear acceleration of the rod's center of mass. Similarly, the sum of the torques will determine the rod's angular acceleration about its center of mass. The motion of an arbitrary point on the rod will be a combination of both motions.
 
  • #3
Is this one of those simple trick questions? I'm thinking that if I put 5N weight on the end of the rod and yank it up by the other end with 6N, it straightens out vertical, and it's like I'm pulling with 1N.
 
  • #4
Farsight said:
Is this one of those simple trick questions? I'm thinking that if I put 5N weight on the end of the rod and yank it up by the other end with 6N, it straightens out vertical, and it's like I'm pulling with 1N.
No.

For one thing, you're assuming the forces are inertial - they point the same direction regardless of the orientation of the rod. The problem doesn't say one way or the other.

If the forces are always perpendicular to the rod, you have 1N of Force causing the center of mass to accelerate. You have 11N * (whatever the distance between the center of mass and the ends) worth of torque causing rotational acceleration.

If the forces are inertial, you have a moving pendulum. The center of mass accelerates due to the 1N force and the rod swings back and forth forever (theoretically, if those are the only two forces that act on the rod).
 

FAQ: F=ma for extended objects and movement of CM

What is the meaning of F=ma for extended objects?

Newton's second law of motion, F=ma, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. For extended objects, this law applies to the movement of its center of mass (CM). In other words, the net force acting on an extended object will cause its CM to accelerate in the direction of the net force.

How is the movement of CM related to an extended object's overall motion?

The movement of an extended object's CM is directly related to its overall motion. If there is no external force acting on the extended object, its CM will remain at rest or move with a constant velocity. However, if there is a net external force acting on the extended object, its CM will accelerate and cause the object to move as a whole.

Can multiple forces act on an extended object's CM?

Yes, multiple forces can act on an extended object's CM. The net force on the CM is the vector sum of all the individual forces acting on it. This means that the CM will accelerate in the direction of the net force, taking into account the magnitudes and directions of all the individual forces.

How does the shape of an extended object affect the movement of its CM?

The shape of an extended object does not affect the movement of its CM as long as the net external force acting on the object is constant. This is because the CM is a point that represents the average position of the object's mass, regardless of its shape. However, if the net force is not constant, the shape of the object can affect the movement of its CM.

Can an extended object rotate and translate at the same time?

Yes, an extended object can rotate and translate simultaneously. This happens when there is a net force acting on the object, causing its CM to accelerate, and a torque acting on the object, causing it to rotate. This is known as a combination of translational and rotational motion, where the CM moves along a straight line while the object rotates around its CM.

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