Q#1
Can the center of mass of a body be at a point outside the body?Answer:
Yes, the center of mass of a body can be outside the body. For example, the center of mass of a semi-circular wire does not lie inside the wire but at a point 2R/π above the center.
If all the particles of a system lie in X-Y plane, is it necessary that the center of mass be in X-Y plane?
Answer:
Yes. Since Z-coordinate of each particle will be zero, hence ഽz.dm = 0. So Z-coordinate of the center of mass Z = (1/M)ഽz.dm = 0. So, the center of mass lies in the X-Y plane.
Q#3
If all the particles of a system lie in a cube, is it necessary that the center of mass be in the cube?
Answer:
Yes. If we start from any two particles of this system, the center of mass (say A) of these two particles will lie on the line joining them. This line will be fully inside the cube. Next, these two particles can be replaced with their total mass at this center of mass A. Now the center of mass of third particle and previous two particles will lie on the line joining the third particle and A, Say this center of mass as B. Again this line joining A and third particle is fully inside the cube, so B is also inside the cube. The total mass of three particles can be assumed to be at B. In this way we can show that the center of mass of all the particles of a system that lies in a cube is necessarily in the cube.
Q#4
The center of mass is defined as R=(1/M)Σmiri. Suppose we define "center of charge" as Rc =(1/Q)Σqiriwhere qi represents the ith charge placed at riand Q is the total charge of the system.
(a) can the center of charge of a two-charge system be outside the line segment joining the charges?
(b) If all the charges of a system are in X-Y plane, is it necessary that the center of charge be in X-Y plane?
(c) If all the charges of a system lie in a cube, is it necessary that the center of charge be in the cube?
Answer:
(a) Yes. Because the charges are of two different type, positive and negative, unlike mass which is all similar type. For example, suppose that there is 2q charge at the origin and -q at 2 units along the x-axis. Its center of charge will be at a distance
= {1/(-q + 2q)}{-q(2) + 2q(0)} = -2q/q = -2
Clearly, the center of charge, in this case, lies outside the line segment joining the two charges.
(b) Since the z-coordinate of all the charges, in this case, is zero so Σqizi = 0.
Hence z-coordinate of center of charge = (1/Q)Σqizi = 0.
It means the center of charge necessarily lies in X -Y plane.
(c) No. As we have seen in (a) that center of charge of two charges may lie outside the line segment joining the two charges, hence the center of charge, in this case, may lie outside the cube.
Q#5
The weight Mg of an extended body is generally shown in a diagram to act through the center of mass. Does it mean that the earth does not attract other particles?
Answer:
The earth does attract each and every particle. The resultant of all these forces of attraction is Mg which acts at the center of mass (where M is the total mass). That is why weight Mg of an extended body is generally shown in a diagram to act through the center of mass.
Q#6
A bob suspended from the ceiling of a car which is accelerating on a horizontal road. The bob stays at rest with respect to the car with the string making an angle θ with the vertical. The linear momentum of the bob as seen from the road is increasing with time. Is it a violation of conservation of linear momentum? If not where is the external force which changes the linear momentum?
Answer:
No, it is not a violation of conservation of linear momentum. The external force is the force of friction of road on the car which is acting through the string of the suspended bob. In the string, it is the tension force, component of which in horizontal direction changes the linear momentum.
Q#7
You are waiting for a train on a railway platform. Your three-year-old niece is standing on your iron trunk containing the luggage. Why does the trunk not recoil as she jumps off on the platform?
Answer:
The force applied by the girl on the trunk is less than the maximum static frictional force on the trunk by the platform.
Q#8
In a head-on collision between two particles, is it necessary that the particles will acquire a common velocity at least for one instant?
Answer:
Yes. Because during the collision the surfaces in contact will move equal distances in equal time interval as long as they remain in contact.
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