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A solid sphere, disc and solid cylinder all of the same mass and made of the same material are allowed to roll down (from rest) on the inclined plane, then
A solid sphere, disc, and solid cylinder of the same material and mass are allowed to roll down an inclined plane from rest. They will have different motions because their moments of inertia are different. The moment of inertia is a measure of how mass is distributed relative to the axis of rotationRead more
A solid sphere, disc, and solid cylinder of the same material and mass are allowed to roll down an inclined plane from rest. They will have different motions because their moments of inertia are different. The moment of inertia is a measure of how mass is distributed relative to the axis of rotation. Among the three shapes, the solid sphere has the smallest moment of inertia in proportion to its mass; hence, the sphere will accelerate faster than the other two down the incline.
As the shapes roll down the slope, they convert gravitational potential energy into kinetic energy. The solid sphere, with its advantageous distribution of mass, will reach the bottom first, followed by the disc. The disc’s moment of inertia is greater than that of the sphere but smaller than that of the solid cylinder, which means it will accelerate more quickly than the cylinder but not as quickly as the sphere.
The solid cylinder, having the largest moment of inertia, will experience the slowest acceleration and therefore will arrive at the bottom of the incline last. Thus, the solid sphere wins the race down the incline, followed by the disc and then the cylinder.
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A circular disc of radius R is removed from a bigger circular disc of radius 2R, such that the circumferences of the disc coincide. The centre of mass of the new disc is αR from the centre of the bigger disc. The value of α is
For finding the value of α as in the extraction of a small disc from the larger disc we take the arrangement, we assume we have large disc with a radius of size 2R and small disc of radius R. After we take out small disc, now we want to know the location of new centre of mass due to this. The largerRead more
For finding the value of α as in the extraction of a small disc from the larger disc we take the arrangement, we assume we have large disc with a radius of size 2R and small disc of radius R. After we take out small disc, now we want to know the location of new centre of mass due to this.
The larger disc has a larger area and mass, so it is initially centered at the origin. The smaller disc that is removed is also centered at the same point. The area and mass of the larger disc are proportionally greater than those of the smaller disc. This difference in mass distribution affects the position of the new center of mass after the smaller disc is removed.
As we analyze the situation, we can infer that the removal of the smaller disc will shift the center of mass toward the larger disc. Applying the principles of mass distribution and balance, we find that the center of mass of the remaining shape is located at a distance of αR from the center of the larger disc. After calculations, we conclude that the value of α is one-third which indicates the relative position of the center of mass in the modified structure.
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A rod of length 1.4 m and negligible mass has two masses of 0.3 kg and 0.7 kg tied to its two ends. Find the location of the point on this rod, where the rotational energy is minimum, when the rod is rotated about the point.
One rod of 1.4 m length has the masses of 0.3 kg and 0.7 kg at both its ends, and its energy is determined while it is in rotational motion with a given rotational point. Let us determine this point for that given rod with respect to which it can have its rotational energy minimized if this rod is sRead more
One rod of 1.4 m length has the masses of 0.3 kg and 0.7 kg at both its ends, and its energy is determined while it is in rotational motion with a given rotational point. Let us determine this point for that given rod with respect to which it can have its rotational energy minimized if this rod is set in rotation at this particular point.
In this case, to find the point of optimal value, one must consider the mass distribution along the rod. In this scenario, the center of mass is of utmost importance, as it denotes the balance point of the system. If the rod rotates about its center of mass, the distances of each mass from that point determine the rotational energy.
The distances of the masses from the point of rotation can directly affect the moment of inertia. It would be perfect to have the axis of rotation closer to the larger mass if it is intended to reduce the rotational energy. Thus, in this configuration, the rotational energy is minimized at a distance of 0.98 meters from the 0.3 kg mass. This setup will make the system function very efficiently, bringing the energy necessary for rotation to a minimum.
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Angular momentum is
Angular momentum is the primary concept that gives a description of rotational motion within any object. The quantity is a vector, and hence it has both magnitude and direction. The angular momentum depends on two factors: moment of inertia and angular velocity. The moment of inertia is a form of quRead more
Angular momentum is the primary concept that gives a description of rotational motion within any object. The quantity is a vector, and hence it has both magnitude and direction. The angular momentum depends on two factors: moment of inertia and angular velocity. The moment of inertia is a form of quantifying the distribution of mass with respect to the axis of rotation of an object, while the angular velocity gives a description of how fast an object is rotating.
In a closed system, the total angular momentum remains unchanged if no net torque is acting upon it. This principle is often termed the law of conservation of angular momentum and is particularly important in the examination of the motion of rotating bodies. For example, if a figure skater pulls her arms in during the spin she performs, her moment of inertia decreases, causing her angular velocity to increase and, hence, keeping the angular momentum unchanged.
There is a calculation of angular momentum for various shapes like disks, spheres, and rigid bodies. It has been very fundamental in understanding the phenomena of planetary motion, for instance, because the angular momentum of celestial objects remains constant when they orbit other larger masses. Angular momentum generally plays a fundamental role in classical and modern physics in terms of dynamics in rotating systems and the universal laws governing the motion.
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A horizontal platform is rotating with uniform angular velocity ω around the vertical axis passing through its centre. At some instant of time, a viscous liquid of mass m is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period.
When a horizontal platform rotates uniformly around a vertical axis passing through its center, the addition of a viscous liquid at its center will affect its motion. After the liquid has been dropped, it spreads outward due to the rotation of the platform and the forces of centrifugation. The movemRead more
When a horizontal platform rotates uniformly around a vertical axis passing through its center, the addition of a viscous liquid at its center will affect its motion. After the liquid has been dropped, it spreads outward due to the rotation of the platform and the forces of centrifugation. The movement of the liquid away from the axis of rotation causes a change in the overall distribution of mass, thus increasing the moment of inertia of the platform.
The principle of conservation of angular momentum tells us that if the torques exerted on a system are zero, the total angular momentum of the system is constant. However, this is dependent both on the moment of inertia and angular velocity, so when the moment of inertia increases due to spreading of liquid, the angular velocity has to reduce for maintaining constant angular momentum. This causes a continuous decrease in the rotation speed of the platform as long as the liquid continues spreading outward.
The angular velocity does not remain constant or increase because the redistribution of mass always increases the moment of inertia. This process ensures that the rotation of the platform slows down uniformly over time, illustrating how angular momentum conservation governs such interactions. Thus, the angular velocity of the platform decreases continuously as the liquid spreads outward and eventually falls off.
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