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Three identical metal balls, each of the radius r are placed touching each other on a horizontal surface such that an equilateral triangle is formed surface such that an equilateral triangle is formed when centres of three balls are joined. The centre of the mass of the system is located at
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the trianglRead more
Three identical metal balls of the same radius are placed on a horizontal surface so that their centers form the vertices of an equilateral triangle. Because all the balls have the same mass and are symmetrically arranged, the center of mass of this system lies at the geometric center of the triangle. This point is known as the centroid of the triangle.
The centroid is a point where all the medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side. In an equilateral triangle, the centroid lies equidistant from all the three vertices and is inside the triangle. That is why it balances the system perfectly due to symmetry as well as uniformity in the mass distribution of the three balls.
Furthermore, since the balls are at rest on that horizontal surface, the center of mass will be on the surface because no part of their mass protrudes vertically either above or below the surface. The center of mass is therefore guaranteed to be in the plane containing the surface. There is an important location associated with the concept of balance and its motion-the centroid. The centroid is essentially the average position of the entire system’s mass.
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See lessA lens of focal length f is splitted into two parts. What is the focal length of either part?
When a lens of focal length f is split into two equal parts along its optical axis, each part retains the same focal length f, as the lens's curvature and refractive power remain unchanged. For more visit here: https://www.tiwariacademy.com/ncert-solutions/class-12/physics/chapter-9/
When a lens of focal length
f is split into two equal parts along its optical axis, each part retains the same focal length
f, as the lens’s curvature and refractive power remain unchanged.
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A rod has length 3 m and its mass acting per unit length is directly proportional to distance x from one of its end, then its centre of gravity from that end will be at
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to tRead more
A rod of length 3 meters has a mass that increases along its length, with the mass per unit length directly proportional to the distance from one end. This means that as you move further along the rod from the starting point, the density of the rod increases, making the far end heavier compared to the starting end.
To find the center of gravity of this rod, we will consider the balance point where the total mass on either side is equal. The rod’s density increases with distance, so its weight is concentrated more toward the far end. Thus, the center of gravity will not be at the midpoint, which is at 1.5 meters, but will shift closer to the heavier end.
By considering the mass distribution and the balance point, it is thus determined that the center of gravity is 2.5 meters from the starting end of the rod. This position then balances the rod just right in account of its mass variability along its length.
Therefore, the center of gravity shifts more towards the denser side, thereby showing how the mass distribution impacts an object’s balancing point.
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See lessA solid sphere of radius R is placed on smooth horizontal surface. A horizontal force F is applied at height h from the lowest point. For the maximum acceleration of centre of mass, which is correct?
When a solid sphere is put on a smooth horizontal plane and a horizontal force is applied, the position where the force has been applied varies with the motion of the ball. The applied force can make both linear motion of the centre of mass and rotational motion about the center. As the result of thRead more
When a solid sphere is put on a smooth horizontal plane and a horizontal force is applied, the position where the force has been applied varies with the motion of the ball. The applied force can make both linear motion of the centre of mass and rotational motion about the center.
As the result of this, the effect of rotation will depend upon the height at which the force has been applied. If the force is applied above the center of the sphere, it creates torque and causes rotation. This decreases the fraction of the force that can be used to accelerate the center of mass linearly. On the other hand, if the force is applied lower on the sphere, closer to its base, the torque is smaller, and a greater fraction of the applied force contributes to linear acceleration.
The linear acceleration of the sphere can be maximized only when the torque is at its minimum. This occurs when the force is applied at the lowest point of the sphere. In this case, the force passes directly through the center of mass of the sphere, meaning there is no rotation and all of the force applied will be utilized for linear acceleration.
Therefore, the maximum acceleration of the center of mass of the sphere occurs when the horizontal force is applied at the lowest point on the sphere.
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See lessThe centre of mass of a system of particles does not depend on
The center of mass of a system of particles is the average position of all the particles, weighted by their masses. It is an intrinsic property of the system and depends on the positions and masses of the individual particles. The position of the center of mass is calculated based on the distributioRead more
The center of mass of a system of particles is the average position of all the particles, weighted by their masses. It is an intrinsic property of the system and depends on the positions and masses of the individual particles. The position of the center of mass is calculated based on the distribution of mass within the system and the relative distances between the particles.
However, it must be stressed that these forces don’t cause an effect in determining the place of the center of mass at a given instant of time. As for a change in position with time for a given overall force acting upon the system, changing positions with external influences, its intrinsic position as for the configuration does not. For example, if a system of particles is exposed to gravity or an external push, then the center moves relative to the center of mass according to the net force. However, its position relative to the system remains unchanged.
The center of mass is determined purely by the spatial arrangement and masses of the particles, independent of any forces on the system, making it a very basic concept in explaining the motion and behavior of physical systems.
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