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.
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.
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.
A wheel rotates from rest at an initial angular speed of 2.00 rad/s with a constant angular acceleration of 3.0 rad/s 2. Over a time interval of 2 s, the total angle through which the wheel rotates can be determined by combining contributions from its initial angular speed and acceleration. The anguRead more
A wheel rotates from rest at an initial angular speed of 2.00 rad/s with a constant angular acceleration of 3.0 rad/s 2. Over a time interval of 2 s, the total angle through which the wheel rotates can be determined by combining contributions from its initial angular speed and acceleration.
The angular displacement depends on two factors: how much the wheel rotates due to its initial speed and how much it accelerates during the given time. The rotation caused by the initial speed is just the product of the angular speed and time, which accounts for a certain number of radians. The additional rotation comes from the angular acceleration, which increases the wheel’s speed over time, causing more rotation.
The total rotation over those 2 s is the sum of these two contributions. In this example, the angle that is rotated solely due to initial velocity is 4 rad; the acceleration adds in an additional 6 rad to the rotation over the same 2 s period. So a total angular displacement of 10 rad would be completed in the 2 s time duration.
This calculation illustrates the combination of the initial angular motion with constant acceleration as leading to an increase in the total angle rotated over time.
For maximum moment of inertia of a circular disc about its geometrical axis, the arrangement of materials comes into play. Moment of inertia depends on the mass of the material and at what distance from the axis of rotation it is located. Given a certain amount of mass, the greater the distance of tRead more
For maximum moment of inertia of a circular disc about its geometrical axis, the arrangement of materials comes into play. Moment of inertia depends on the mass of the material and at what distance from the axis of rotation it is located. Given a certain amount of mass, the greater the distance of that mass from the axis of rotation, the greater the corresponding contribution to the moment of inertia.
In this case, the disc is made using iron and aluminum. Iron is denser and heavier, while aluminum is lighter. The heavier material should be placed farther from the axis of rotation to get maximum moment of inertia. The moment of inertia increases with the square of the distance from the axis. By placing iron in the outer region of the disc and aluminum closer to the center, the heavier material contributes more effectively to the rotational resistance.
This arrangement ensures the mass farther away from the axis maximizes its contribution to the moment of inertia. Alternatively, placing aluminum on the interior will reduce its less significant contribution to the inertia. Therefore, the optimal setup is to place aluminum at the interior and iron to surround it, thus maximizing moment of inertia.
The moment of inertia of a uniform circular disc depends on the axis of rotation. Considering an axis passing through its center and perpendicular to its plane, the opposition to rotation is minimal compared with when the axis shifts to another location. If the axis is changed so that it is moved toRead more
The moment of inertia of a uniform circular disc depends on the axis of rotation. Considering an axis passing through its center and perpendicular to its plane, the opposition to rotation is minimal compared with when the axis shifts to another location. If the axis is changed so that it is moved to a point on the rim of the disc but remains perpendicular to the plane, the moment of inertia will increase.
This increases because the mass is now farther away from the new axis, making it harder for the disc to rotate. To determine this new moment of inertia, we can use the concept of adding the effect of the shifting axis. This additional factor accounts for how much the mass is distributed away from the original central axis.
The final outcome is that the moment of inertia about the rim is five times the moment of inertia about a diameter of the disc. This gives an idea about how moving the axis away from the center increases rotational resistance significantly. It is quite an important concept in mechanics for predicting the behavior of objects under different rotational conditions and for the design of systems involving rotating components.
A flywheel is rotating about a fixed axis with a kinetic energy of 360 joules and an angular speed of 30 radians per second. Determine the moment of inertia of the flywheel about its axis of rotation. The moment of inertia is a measure of how mass is distributed in a rotating object and how difficulRead more
A flywheel is rotating about a fixed axis with a kinetic energy of 360 joules and an angular speed of 30 radians per second. Determine the moment of inertia of the flywheel about its axis of rotation.
The moment of inertia is a measure of how mass is distributed in a rotating object and how difficult it is to change the rotational motion of that object. It plays a role in rotational dynamics just like the role played by mass in linear motion. When the angular speed and the moment of inertia are known, then the kinetic energy of the rotating object can be calculated.
The calculations for a given flywheel will show 0.8 kg·m² to be its moment of inertia, meaning its distribution of mass and rotational resistance matches this quantity. Understanding moments of inertia supports designing and even analyzing systems like an engine, turbines, or any mechanical flywheel for efficient safety and successful operation in general.
Thus, the moment of inertia of the flywheel is highly indicative of how it could store rotational energy and resist changes in its motion.
The time period of a planet's revolution around the Sun is governed by Kepler's Third Law, which establishes a relationship between the orbital period and the distance of the planet from the Sun. According to this law, the square of the orbital period of a planet is proportional to the cube of its aRead more
The time period of a planet’s revolution around the Sun is governed by Kepler’s Third Law, which establishes a relationship between the orbital period and the distance of the planet from the Sun. According to this law, the square of the orbital period of a planet is proportional to the cube of its average distance from the Sun. This means that as the distance of a planet from the Sun increases, its orbital period becomes significantly longer.
In this scenario, the distances of two planets from the Sun are 10¹³ meters and 10¹² meters, respectively. The ratio of their orbital periods can be determined using Kepler’s Third Law. For the first planet, which is farther from the Sun, the time period increases because the gravitational pull decreases with distance, resulting in a slower orbital speed.
Using the law, it is found that the ratio of the time periods of the two planets is 10√10. This value shows that the first planet, being ten times farther from the Sun, takes considerably longer to complete one revolution than the second planet. This result demonstrates the profound effect of distance on the orbital dynamics of celestial bodies in a solar system.
When a body of mass m is moved from the surface of the Earth to a height equal to three times the Earth's radius h = 3R, there is a change in its gravitational potential energy. Gravitational potential energy depends on the position of the body relative to the center of the Earth and decreases as thRead more
When a body of mass m is moved from the surface of the Earth to a height equal to three times the Earth’s radius h = 3R, there is a change in its gravitational potential energy. Gravitational potential energy depends on the position of the body relative to the center of the Earth and decreases as the distance from the center increases.
At the surface of the Earth, the body’s potential energy is determined by the distance R from the center. When the body is taken to a height of h = 3R, the total distance from the Earth’s center becomes 4R. The gravitational potential energy at these two points differs because potential energy is inversely proportional to the distance from the center of the Earth.
The change in gravitational potential energy is calculated as the difference between the potential energy at the surface and at the height h = 3R. After simplifying the relationship, it is found that the change in potential energy is mgR/4.
This result reflects how gravitational potential energy decreases with increasing distance from the center of the Earth. It also demonstrates the significance of height and mass in calculating energy changes during such movements, essential in space travel and satellite deployment.
The escape velocity is defined as the minimum speed by which a body has to move away from the influence of a planet's gravitational pull without further propulsion. For Earth, it would depend on how far a distance one moves away from the center of the Earth. The escape velocity is proportional to thRead more
The escape velocity is defined as the minimum speed by which a body has to move away from the influence of a planet’s gravitational pull without further propulsion. For Earth, it would depend on how far a distance one moves away from the center of the Earth. The escape velocity is proportional to the square root of the reciprocal of the radius R of the Earth at the surface.
Now consider a platform located at a height equal to the radius of the Earth (R) above its surface. This makes the total distance from the Earth’s center to the platform (2R). Since escape velocity decreases with an increase in distance from the planet’s center, the escape velocity from this platform is less than that from the Earth’s surface.
This, knowing that escape velocity varies in inverse proportion to the distance from the center’s square root, makes escape velocity at a distance of 2R a fraction of the surface escape velocity. The factor f linking between the two velocities can then be calculated as follows, 1/√2 . Therefore, if the escape velocity from earth’s surface is some quantity ‘a’, then from this ‘platform’ it becomes an amount ‘a divided by f’.
This shows that gravity depends on distance, indicating that the escape velocity at altitude varies.
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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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
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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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
A wheel has angular acceleration of 3.0 rad/sec² and an initial angular speed of 2.00 rad/sec². In a time of 2 sec, it has rotated through an angle (in radian) of
A wheel rotates from rest at an initial angular speed of 2.00 rad/s with a constant angular acceleration of 3.0 rad/s 2. Over a time interval of 2 s, the total angle through which the wheel rotates can be determined by combining contributions from its initial angular speed and acceleration. The anguRead more
A wheel rotates from rest at an initial angular speed of 2.00 rad/s with a constant angular acceleration of 3.0 rad/s 2. Over a time interval of 2 s, the total angle through which the wheel rotates can be determined by combining contributions from its initial angular speed and acceleration.
The angular displacement depends on two factors: how much the wheel rotates due to its initial speed and how much it accelerates during the given time. The rotation caused by the initial speed is just the product of the angular speed and time, which accounts for a certain number of radians. The additional rotation comes from the angular acceleration, which increases the wheel’s speed over time, causing more rotation.
The total rotation over those 2 s is the sum of these two contributions. In this example, the angle that is rotated solely due to initial velocity is 4 rad; the acceleration adds in an additional 6 rad to the rotation over the same 2 s period. So a total angular displacement of 10 rad would be completed in the 2 s time duration.
This calculation illustrates the combination of the initial angular motion with constant acceleration as leading to an increase in the total angle rotated over time.
Click here :https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessA circular disc is to be made by using iron and aluminium so that it acquired maximum moment of inertia about geometrical axis. It is possible with
For maximum moment of inertia of a circular disc about its geometrical axis, the arrangement of materials comes into play. Moment of inertia depends on the mass of the material and at what distance from the axis of rotation it is located. Given a certain amount of mass, the greater the distance of tRead more
For maximum moment of inertia of a circular disc about its geometrical axis, the arrangement of materials comes into play. Moment of inertia depends on the mass of the material and at what distance from the axis of rotation it is located. Given a certain amount of mass, the greater the distance of that mass from the axis of rotation, the greater the corresponding contribution to the moment of inertia.
In this case, the disc is made using iron and aluminum. Iron is denser and heavier, while aluminum is lighter. The heavier material should be placed farther from the axis of rotation to get maximum moment of inertia. The moment of inertia increases with the square of the distance from the axis. By placing iron in the outer region of the disc and aluminum closer to the center, the heavier material contributes more effectively to the rotational resistance.
This arrangement ensures the mass farther away from the axis maximizes its contribution to the moment of inertia. Alternatively, placing aluminum on the interior will reduce its less significant contribution to the inertia. Therefore, the optimal setup is to place aluminum at the interior and iron to surround it, thus maximizing moment of inertia.
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See lessMoment of inertia of a uniform circular disc about a diameter is l. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be
The moment of inertia of a uniform circular disc depends on the axis of rotation. Considering an axis passing through its center and perpendicular to its plane, the opposition to rotation is minimal compared with when the axis shifts to another location. If the axis is changed so that it is moved toRead more
The moment of inertia of a uniform circular disc depends on the axis of rotation. Considering an axis passing through its center and perpendicular to its plane, the opposition to rotation is minimal compared with when the axis shifts to another location. If the axis is changed so that it is moved to a point on the rim of the disc but remains perpendicular to the plane, the moment of inertia will increase.
This increases because the mass is now farther away from the new axis, making it harder for the disc to rotate. To determine this new moment of inertia, we can use the concept of adding the effect of the shifting axis. This additional factor accounts for how much the mass is distributed away from the original central axis.
The final outcome is that the moment of inertia about the rim is five times the moment of inertia about a diameter of the disc. This gives an idea about how moving the axis away from the center increases rotational resistance significantly. It is quite an important concept in mechanics for predicting the behavior of objects under different rotational conditions and for the design of systems involving rotating components.
Click here for NCERT Solutions:- https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessA flywheel rotating about fixed axis has a kinetic energy of 360 joule when its angular speed is 30 radian/sec. The moment of inertia of the wheel about the axis of rotation is
A flywheel is rotating about a fixed axis with a kinetic energy of 360 joules and an angular speed of 30 radians per second. Determine the moment of inertia of the flywheel about its axis of rotation. The moment of inertia is a measure of how mass is distributed in a rotating object and how difficulRead more
A flywheel is rotating about a fixed axis with a kinetic energy of 360 joules and an angular speed of 30 radians per second. Determine the moment of inertia of the flywheel about its axis of rotation.
The moment of inertia is a measure of how mass is distributed in a rotating object and how difficult it is to change the rotational motion of that object. It plays a role in rotational dynamics just like the role played by mass in linear motion. When the angular speed and the moment of inertia are known, then the kinetic energy of the rotating object can be calculated.
The calculations for a given flywheel will show 0.8 kg·m² to be its moment of inertia, meaning its distribution of mass and rotational resistance matches this quantity. Understanding moments of inertia supports designing and even analyzing systems like an engine, turbines, or any mechanical flywheel for efficient safety and successful operation in general.
Thus, the moment of inertia of the flywheel is highly indicative of how it could store rotational energy and resist changes in its motion.
Click here for more:- https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessThe distance of two planets from the sun are 10¹³ m and 10¹² m respectively. The ratio of time periods of the planets is
The time period of a planet's revolution around the Sun is governed by Kepler's Third Law, which establishes a relationship between the orbital period and the distance of the planet from the Sun. According to this law, the square of the orbital period of a planet is proportional to the cube of its aRead more
The time period of a planet’s revolution around the Sun is governed by Kepler’s Third Law, which establishes a relationship between the orbital period and the distance of the planet from the Sun. According to this law, the square of the orbital period of a planet is proportional to the cube of its average distance from the Sun. This means that as the distance of a planet from the Sun increases, its orbital period becomes significantly longer.
In this scenario, the distances of two planets from the Sun are 10¹³ meters and 10¹² meters, respectively. The ratio of their orbital periods can be determined using Kepler’s Third Law. For the first planet, which is farther from the Sun, the time period increases because the gravitational pull decreases with distance, resulting in a slower orbital speed.
Using the law, it is found that the ratio of the time periods of the two planets is 10√10. This value shows that the first planet, being ten times farther from the Sun, takes considerably longer to complete one revolution than the second planet. This result demonstrates the profound effect of distance on the orbital dynamics of celestial bodies in a solar system.
See lessA body of mass m is placed on earth surface which is taken from earth surface to a height of h = 3 R. Then change in gravitational
When a body of mass m is moved from the surface of the Earth to a height equal to three times the Earth's radius h = 3R, there is a change in its gravitational potential energy. Gravitational potential energy depends on the position of the body relative to the center of the Earth and decreases as thRead more
When a body of mass m is moved from the surface of the Earth to a height equal to three times the Earth’s radius h = 3R, there is a change in its gravitational potential energy. Gravitational potential energy depends on the position of the body relative to the center of the Earth and decreases as the distance from the center increases.
At the surface of the Earth, the body’s potential energy is determined by the distance R from the center. When the body is taken to a height of h = 3R, the total distance from the Earth’s center becomes 4R. The gravitational potential energy at these two points differs because potential energy is inversely proportional to the distance from the center of the Earth.
The change in gravitational potential energy is calculated as the difference between the potential energy at the surface and at the height h = 3R. After simplifying the relationship, it is found that the change in potential energy is mgR/4.
This result reflects how gravitational potential energy decreases with increasing distance from the center of the Earth. It also demonstrates the significance of height and mass in calculating energy changes during such movements, essential in space travel and satellite deployment.
See lessThe earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the earth. The escape velocity of a body from this platform is fv, where v is its escape velocity from the surface of the earth. The value of f is
The escape velocity is defined as the minimum speed by which a body has to move away from the influence of a planet's gravitational pull without further propulsion. For Earth, it would depend on how far a distance one moves away from the center of the Earth. The escape velocity is proportional to thRead more
The escape velocity is defined as the minimum speed by which a body has to move away from the influence of a planet’s gravitational pull without further propulsion. For Earth, it would depend on how far a distance one moves away from the center of the Earth. The escape velocity is proportional to the square root of the reciprocal of the radius R of the Earth at the surface.
Now consider a platform located at a height equal to the radius of the Earth (R) above its surface. This makes the total distance from the Earth’s center to the platform (2R). Since escape velocity decreases with an increase in distance from the planet’s center, the escape velocity from this platform is less than that from the Earth’s surface.
This, knowing that escape velocity varies in inverse proportion to the distance from the center’s square root, makes escape velocity at a distance of 2R a fraction of the surface escape velocity. The factor f linking between the two velocities can then be calculated as follows, 1/√2 . Therefore, if the escape velocity from earth’s surface is some quantity ‘a’, then from this ‘platform’ it becomes an amount ‘a divided by f’.
This shows that gravity depends on distance, indicating that the escape velocity at altitude varies.
See less