When a person standing on a rotating disc stretches out his hands, his angular speed decreases. This is because of the law of conservation of angular momentum, where the angular momentum of a system remains the same if no external torque acts on it. Angular momentum is the product of the moment of iRead more
When a person standing on a rotating disc stretches out his hands, his angular speed decreases. This is because of the law of conservation of angular momentum, where the angular momentum of a system remains the same if no external torque acts on it.
Angular momentum is the product of the moment of inertia and angular velocity. If he stretches his arms out to his sides, the person increases his moment of inertia, as the mass distribution moves farther from the axis of rotation. In order to conserve angular momentum, the angular velocity-or angular speed-must decrease correspondingly.
This principle is often seen in figure skating or gymnastics. A spinning skater can raise his speed by retracting his arms so that he reduces his moment of inertia and increases his angular velocity. He slows down if he stretches out his arms.
This principle also applies to many real-world circumstances, such as athletes using body movements to control rotational speed or space probes adjusting the orientation of their trajectory in space. In such a case, by extending their arms, the moment of inertia is increased, leading to a decrease in angular velocity to keep angular momentum constant.
Total kinetic energy, in this case of a rolling sphere, consists of two parts: rotational kinetic energy and translational kinetic energy. Rotational kinetic energy is that caused by rotation of the sphere around its axis of rotation while the translational kinetic energy arises due to linear motionRead more
Total kinetic energy, in this case of a rolling sphere, consists of two parts: rotational kinetic energy and translational kinetic energy. Rotational kinetic energy is that caused by rotation of the sphere around its axis of rotation while the translational kinetic energy arises due to linear motion of the sphere. In rolling motion, these two forms of energy are coupled through the condition that the sphere rolls without sliding, implying there is no relative motion at the point of contact with the surface.
Rotational kinetic energy for a solid sphere is a particular fraction of total kinetic energy. For the given problem, the rotational kinetic energy: total kinetic energy ratio was 2/7. This suggests that the rotational part contributes less to the total energy as compared to the translational part. The remaining 5/7 is all taken up by the translation kinetic energy.
This concept is important in understanding how energy is distributed in rolling systems. It has practical applications in fields like mechanics, engineering, and physics education, helping to analyze the motion of rolling objects such as balls, wheels, and gears. The ratio emphasizes the importance of both rotational and translational dynamics in rolling motion.
The moment of inertia is a property that tells us how a rotating object resists the change in angular acceleration. The theoretical background assumes mass, the distribution of this mass relative to the axis, and the selected axis itself for the three parameters. The mass and how far away from the aRead more
The moment of inertia is a property that tells us how a rotating object resists the change in angular acceleration. The theoretical background assumes mass, the distribution of this mass relative to the axis, and the selected axis itself for the three parameters. The mass and how far away from the axis it lies give an object a higher moment of inertia. For instance, a hollow ring has a greater moment of inertia than an equivalent solid disc of the same mass and radius because its mass is distributed further from the axis. However, the moment of inertia is independent of the angular velocity of the object. Angular velocity explains how fast an object is spinning, but this does not contribute to the intrinsic resistance to its rotation. It means that irrespective of whether it is spinning rapidly or slowly, the moment of inertia remains unchanged because it only depends on its physical structure and the axis about which it rotates.
Determine the ratio of moments of inertia between a circular ring and a circular disc with equal masses and radius using a comparison of their rotation behavior about an axis passing through the center and perpendicular to the planes of the bodies in question. The moment of inertia of a circular rinRead more
Determine the ratio of moments of inertia between a circular ring and a circular disc with equal masses and radius using a comparison of their rotation behavior about an axis passing through the center and perpendicular to the planes of the bodies in question.
The moment of inertia of a circular ring depends entirely on its mass being concentrated at its outer edge. For a ring, all the mass is at the same radius, making its moment of inertia proportional to the square of the radius multiplied by its mass.
On the other hand, a circular disc has all of its mass distributed along its entire surface. Some of its masses are closer to the axis of rotation than if all the mass had been concentrated to the outer edge of the disc. Thus its moment of inertia will be lower compared to that of the ring. The result of the moment of inertia in a disc comes from considering its mass distribution:.
Comparing both, the moment of inertia of the ring is twice that of the disc for the same mass and radius. Thus, the ratio of their moments of inertia is 2:1, with the ring having the larger value.
The moments of inertia of two rings, one with radius r and the other with radius nr, can be compared when studying their rotational dynamics about an axis perpendicular to their planes and passing through their centers. The moment of inertia is a measure of an object's resistance to angular acceleraRead more
The moments of inertia of two rings, one with radius r and the other with radius nr, can be compared when studying their rotational dynamics about an axis perpendicular to their planes and passing through their centers. The moment of inertia is a measure of an object’s resistance to angular acceleration when a torque is applied. For rings, mass and the square of radius can be used to calculate moments of inertia, so in that case, for our first ring: the radius is taken as r . For the second ring, in this case a multiple of first, its radius is taken to be nr.
When the moments of inertia for each ring are calculated, the ring with radius nr will have a moment of inertia proportional to the square of its radius compared to the first ring. Hence, if we find the ratio of their moments of inertia, we would get that the moment of inertia for the second ring is proportional to the square of the scaling factor n . Thus, the ratio of the moments of inertia of the two rings comes out to be 1 : n² . This also explains how mass distribution and geometry influence rotational motion.
If a person standing on a rotating disc stretches out his hands, the angular speed will
When a person standing on a rotating disc stretches out his hands, his angular speed decreases. This is because of the law of conservation of angular momentum, where the angular momentum of a system remains the same if no external torque acts on it. Angular momentum is the product of the moment of iRead more
When a person standing on a rotating disc stretches out his hands, his angular speed decreases. This is because of the law of conservation of angular momentum, where the angular momentum of a system remains the same if no external torque acts on it.
Angular momentum is the product of the moment of inertia and angular velocity. If he stretches his arms out to his sides, the person increases his moment of inertia, as the mass distribution moves farther from the axis of rotation. In order to conserve angular momentum, the angular velocity-or angular speed-must decrease correspondingly.
This principle is often seen in figure skating or gymnastics. A spinning skater can raise his speed by retracting his arms so that he reduces his moment of inertia and increases his angular velocity. He slows down if he stretches out his arms.
This principle also applies to many real-world circumstances, such as athletes using body movements to control rotational speed or space probes adjusting the orientation of their trajectory in space. In such a case, by extending their arms, the moment of inertia is increased, leading to a decrease in angular velocity to keep angular momentum constant.
See more : – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessA sphere of radius r is rolling without sliding. What is ratio of rotational kinetic energy and total kinetic energy associated with the sphere?
Total kinetic energy, in this case of a rolling sphere, consists of two parts: rotational kinetic energy and translational kinetic energy. Rotational kinetic energy is that caused by rotation of the sphere around its axis of rotation while the translational kinetic energy arises due to linear motionRead more
Total kinetic energy, in this case of a rolling sphere, consists of two parts: rotational kinetic energy and translational kinetic energy. Rotational kinetic energy is that caused by rotation of the sphere around its axis of rotation while the translational kinetic energy arises due to linear motion of the sphere. In rolling motion, these two forms of energy are coupled through the condition that the sphere rolls without sliding, implying there is no relative motion at the point of contact with the surface.
Rotational kinetic energy for a solid sphere is a particular fraction of total kinetic energy. For the given problem, the rotational kinetic energy: total kinetic energy ratio was 2/7. This suggests that the rotational part contributes less to the total energy as compared to the translational part. The remaining 5/7 is all taken up by the translation kinetic energy.
This concept is important in understanding how energy is distributed in rolling systems. It has practical applications in fields like mechanics, engineering, and physics education, helping to analyze the motion of rolling objects such as balls, wheels, and gears. The ratio emphasizes the importance of both rotational and translational dynamics in rolling motion.
Click here for more info: – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessMoment of inertia of an object does not depend upon
The moment of inertia is a property that tells us how a rotating object resists the change in angular acceleration. The theoretical background assumes mass, the distribution of this mass relative to the axis, and the selected axis itself for the three parameters. The mass and how far away from the aRead more
The moment of inertia is a property that tells us how a rotating object resists the change in angular acceleration. The theoretical background assumes mass, the distribution of this mass relative to the axis, and the selected axis itself for the three parameters. The mass and how far away from the axis it lies give an object a higher moment of inertia. For instance, a hollow ring has a greater moment of inertia than an equivalent solid disc of the same mass and radius because its mass is distributed further from the axis. However, the moment of inertia is independent of the angular velocity of the object. Angular velocity explains how fast an object is spinning, but this does not contribute to the intrinsic resistance to its rotation. It means that irrespective of whether it is spinning rapidly or slowly, the moment of inertia remains unchanged because it only depends on its physical structure and the axis about which it rotates.
See more:- https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessOne circular ring and one circular disc both are having the same mass and radius. The ratio of their moments of inertia about the axis passing through their centre and perpendicular to their planes will be
Determine the ratio of moments of inertia between a circular ring and a circular disc with equal masses and radius using a comparison of their rotation behavior about an axis passing through the center and perpendicular to the planes of the bodies in question. The moment of inertia of a circular rinRead more
Determine the ratio of moments of inertia between a circular ring and a circular disc with equal masses and radius using a comparison of their rotation behavior about an axis passing through the center and perpendicular to the planes of the bodies in question.
The moment of inertia of a circular ring depends entirely on its mass being concentrated at its outer edge. For a ring, all the mass is at the same radius, making its moment of inertia proportional to the square of the radius multiplied by its mass.
On the other hand, a circular disc has all of its mass distributed along its entire surface. Some of its masses are closer to the axis of rotation than if all the mass had been concentrated to the outer edge of the disc. Thus its moment of inertia will be lower compared to that of the ring. The result of the moment of inertia in a disc comes from considering its mass distribution:.
Comparing both, the moment of inertia of the ring is twice that of the disc for the same mass and radius. Thus, the ratio of their moments of inertia is 2:1, with the ring having the larger value.
Checkout for more info: – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessWhat is the ratio of the moments of inertia of two rings of radii r and nr about an axis perpendicular to their plane and passing through their centers?
The moments of inertia of two rings, one with radius r and the other with radius nr, can be compared when studying their rotational dynamics about an axis perpendicular to their planes and passing through their centers. The moment of inertia is a measure of an object's resistance to angular acceleraRead more
The moments of inertia of two rings, one with radius r and the other with radius nr, can be compared when studying their rotational dynamics about an axis perpendicular to their planes and passing through their centers. The moment of inertia is a measure of an object’s resistance to angular acceleration when a torque is applied. For rings, mass and the square of radius can be used to calculate moments of inertia, so in that case, for our first ring: the radius is taken as r . For the second ring, in this case a multiple of first, its radius is taken to be nr.
When the moments of inertia for each ring are calculated, the ring with radius nr will have a moment of inertia proportional to the square of its radius compared to the first ring. Hence, if we find the ratio of their moments of inertia, we would get that the moment of inertia for the second ring is proportional to the square of the scaling factor n . Thus, the ratio of the moments of inertia of the two rings comes out to be 1 : n² . This also explains how mass distribution and geometry influence rotational motion.
Click here : – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See less