The kinetic energy of a rolling body is a combination of translational and rotational. An object that rolls has a linear velocity at its center of mass, as well as rotation about that center. Translational kinetic energy depends on the mass of the object and its velocity. This energy is associated wRead more
The kinetic energy of a rolling body is a combination of translational and rotational. An object that rolls has a linear velocity at its center of mass, as well as rotation about that center. Translational kinetic energy depends on the mass of the object and its velocity. This energy is associated with the motion of the entire body through space.
In addition to translational kinetic energy, the object has rotational kinetic energy because it is rotating about its axis. The amount of rotational energy depends on the moment of inertia of the object, which is a function of the mass distribution and shape of the object and its angular velocity, which is a measure of how fast it is rotating.
When an object rolls without slipping, there is a relationship between its linear velocity and angular velocity. Specifically, the center of mass velocity is directly proportional to the angular velocity of the object. This relationship forms the basis of the understanding for the conservation of energy in rolling motion. Consequently, the total kinetic energy of a rolling body is the sum of its translational and rotational energies, reflecting the movement through space and the rotation about its axis. This total energy is vital in the analysis of the dynamics of rolling objects within different physical settings.
A mass is tied to a thin, light string that is wound around a cylinder. The mass is released. There is a gravitational force pulling it downward, and the tension in the string is pulling it upward. This difference between the gravitational force and the tension can be calculated to find the net forcRead more
A mass is tied to a thin, light string that is wound around a cylinder. The mass is released. There is a gravitational force pulling it downward, and the tension in the string is pulling it upward. This difference between the gravitational force and the tension can be calculated to find the net force on the mass. By Newton’s second law, this net force must equal the product of the mass and its linear acceleration.
There’s also tension in the string creating a torque in the cylinder which can be represented in terms of tension and radius of the cylinder. The angular acceleration is found to be associated with the linear acceleration of mass by the radius of the cylinder.
We can obtain the linear acceleration of the falling mass and the tension in the string by considering these relationships. The acceleration can be written as a fraction of the acceleration due to gravity, showing that it is smaller than the acceleration due to gravity. This happens because part of the gravitational force is being utilized to generate rotational motion in the cylinder, meaning that there will be a smaller linear acceleration of the mass than if it were falling freely. This whole system demonstrates principles of rotational dynamics and linear motion.
In finding the linear acceleration of a rolling cylinder on an inclined plane, we have the forces involved-the weight of the cylinder and the frictional force. The weight can be divided into two: one along the incline and the other perpendicular to it. The friction force opposes the motion to avoidRead more
In finding the linear acceleration of a rolling cylinder on an inclined plane, we have the forces involved-the weight of the cylinder and the frictional force. The weight can be divided into two: one along the incline and the other perpendicular to it. The friction force opposes the motion to avoid sliding.
Using Newton’s second law, we analyze the net force acting along the incline on the cylinder. The net force is the difference between the component of gravitational force pulling it down the slope and the frictional force acting up the slope. This net force is also related to the linear acceleration of the cylinder.
Next, we analyze the cylinder’s rotational motion where the frictional force causes a torque around the center of mass. Torque affects the cylinder’s angular acceleration. Linear acceleration and angular acceleration for rolling without slipping are in fact related.
Integrating these considerations shows that the linear acceleration of the rolling cylinder down the incline is a fraction of gravitational acceleration. The no slipping requirement for the rolling cylinder further depends on the coefficient of static friction and the angle of the incline so that the frictional force is always quite sufficient to prevent slipping.
When a uniform square plate rotates about an axis in its plane, the moment of inertia depends on the orientation of the axis relative to the sides of the plate. Given that the moment of inertia about an axis parallel to two sides is denoted as l, and the axis CD makes an angle θ with this axis, theRead more
When a uniform square plate rotates about an axis in its plane, the moment of inertia depends on the orientation of the axis relative to the sides of the plate. Given that the moment of inertia about an axis parallel to two sides is denoted as l, and the axis CD makes an angle θ with this axis, the moment of inertia about axis CD is l/cos² θ.
This is because the rotational inertia depends on the distribution of mass relative to the axis of rotation. The cosine squared term accounts for the projection of the mass distribution onto the new axis.
Thus, the correct answer is that the moment of inertia about axis CD is l /cos² θ.
If the ball were to strike the floor, hit the ground and rebound as in an inelastic collision, it would follow that the total momentum of the Earth and the ball is conserved. However, in the inelastic collision, some of its kinetic energy becomes other types of energy like heat or sound that causesRead more
If the ball were to strike the floor, hit the ground and rebound as in an inelastic collision, it would follow that the total momentum of the Earth and the ball is conserved. However, in the inelastic collision, some of its kinetic energy becomes other types of energy like heat or sound that causes loss of mechanical energy but the system with both Earth and the ball together conserves their total momentum.
This is a basic principle in physics, which states that the total momentum of a closed system remains constant if no external forces act upon it. In this case, the ball and the Earth constitute a closed system, and the internal forces during the collision do not affect the total momentum.
Note that, although momentum is conserved, mechanical energy is not in inelastic collisions because kinetic energy is converted into other forms of energy. Thus, the correct answer is that the total momentum of the ball and the Earth is conserved.
Write an expression for the kinetic energy of a body rolling without slipping.
The kinetic energy of a rolling body is a combination of translational and rotational. An object that rolls has a linear velocity at its center of mass, as well as rotation about that center. Translational kinetic energy depends on the mass of the object and its velocity. This energy is associated wRead more
The kinetic energy of a rolling body is a combination of translational and rotational. An object that rolls has a linear velocity at its center of mass, as well as rotation about that center. Translational kinetic energy depends on the mass of the object and its velocity. This energy is associated with the motion of the entire body through space.
In addition to translational kinetic energy, the object has rotational kinetic energy because it is rotating about its axis. The amount of rotational energy depends on the moment of inertia of the object, which is a function of the mass distribution and shape of the object and its angular velocity, which is a measure of how fast it is rotating.
When an object rolls without slipping, there is a relationship between its linear velocity and angular velocity. Specifically, the center of mass velocity is directly proportional to the angular velocity of the object. This relationship forms the basis of the understanding for the conservation of energy in rolling motion. Consequently, the total kinetic energy of a rolling body is the sum of its translational and rotational energies, reflecting the movement through space and the rotation about its axis. This total energy is vital in the analysis of the dynamics of rolling objects within different physical settings.
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See lessA light string is wound round a cylinder and carries a mass tied to it at the free end. When the mass is released, calculate (a) the linear acceleration of the descending mass and (b) the angular acceleration of the cylinder and (c) the tension in the string. Show that the acceleration of mass is less than g.
A mass is tied to a thin, light string that is wound around a cylinder. The mass is released. There is a gravitational force pulling it downward, and the tension in the string is pulling it upward. This difference between the gravitational force and the tension can be calculated to find the net forcRead more
A mass is tied to a thin, light string that is wound around a cylinder. The mass is released. There is a gravitational force pulling it downward, and the tension in the string is pulling it upward. This difference between the gravitational force and the tension can be calculated to find the net force on the mass. By Newton’s second law, this net force must equal the product of the mass and its linear acceleration.
There’s also tension in the string creating a torque in the cylinder which can be represented in terms of tension and radius of the cylinder. The angular acceleration is found to be associated with the linear acceleration of mass by the radius of the cylinder.
We can obtain the linear acceleration of the falling mass and the tension in the string by considering these relationships. The acceleration can be written as a fraction of the acceleration due to gravity, showing that it is smaller than the acceleration due to gravity. This happens because part of the gravitational force is being utilized to generate rotational motion in the cylinder, meaning that there will be a smaller linear acceleration of the mass than if it were falling freely. This whole system demonstrates principles of rotational dynamics and linear motion.
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See lessObtain the expression for the linear acceleration of a cylinder rolling down inclined place and hence find the condition for the cylinder to roll down without slipping.
In finding the linear acceleration of a rolling cylinder on an inclined plane, we have the forces involved-the weight of the cylinder and the frictional force. The weight can be divided into two: one along the incline and the other perpendicular to it. The friction force opposes the motion to avoidRead more
In finding the linear acceleration of a rolling cylinder on an inclined plane, we have the forces involved-the weight of the cylinder and the frictional force. The weight can be divided into two: one along the incline and the other perpendicular to it. The friction force opposes the motion to avoid sliding.
Using Newton’s second law, we analyze the net force acting along the incline on the cylinder. The net force is the difference between the component of gravitational force pulling it down the slope and the frictional force acting up the slope. This net force is also related to the linear acceleration of the cylinder.
Next, we analyze the cylinder’s rotational motion where the frictional force causes a torque around the center of mass. Torque affects the cylinder’s angular acceleration. Linear acceleration and angular acceleration for rolling without slipping are in fact related.
Integrating these considerations shows that the linear acceleration of the rolling cylinder down the incline is a fraction of gravitational acceleration. The no slipping requirement for the rolling cylinder further depends on the coefficient of static friction and the angle of the incline so that the frictional force is always quite sufficient to prevent slipping.
see more :- https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See lessLet l be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle θ with AB. The moment of inertia of the plate about the axis CD is then equal to
When a uniform square plate rotates about an axis in its plane, the moment of inertia depends on the orientation of the axis relative to the sides of the plate. Given that the moment of inertia about an axis parallel to two sides is denoted as l, and the axis CD makes an angle θ with this axis, theRead more
When a uniform square plate rotates about an axis in its plane, the moment of inertia depends on the orientation of the axis relative to the sides of the plate. Given that the moment of inertia about an axis parallel to two sides is denoted as l, and the axis CD makes an angle θ with this axis, the moment of inertia about axis CD is l/cos² θ.
This is because the rotational inertia depends on the distribution of mass relative to the axis of rotation. The cosine squared term accounts for the projection of the mass distribution onto the new axis.
Thus, the correct answer is that the moment of inertia about axis CD is l /cos² θ.
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See lessA ball hits the floor and rebounds after an inelastic collision. In this case
If the ball were to strike the floor, hit the ground and rebound as in an inelastic collision, it would follow that the total momentum of the Earth and the ball is conserved. However, in the inelastic collision, some of its kinetic energy becomes other types of energy like heat or sound that causesRead more
If the ball were to strike the floor, hit the ground and rebound as in an inelastic collision, it would follow that the total momentum of the Earth and the ball is conserved. However, in the inelastic collision, some of its kinetic energy becomes other types of energy like heat or sound that causes loss of mechanical energy but the system with both Earth and the ball together conserves their total momentum.
This is a basic principle in physics, which states that the total momentum of a closed system remains constant if no external forces act upon it. In this case, the ball and the Earth constitute a closed system, and the internal forces during the collision do not affect the total momentum.
Note that, although momentum is conserved, mechanical energy is not in inelastic collisions because kinetic energy is converted into other forms of energy. Thus, the correct answer is that the total momentum of the ball and the Earth is conserved.
See more : – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See less