Rotational Motion

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical Then the angular velocity in this position is:-

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical Force applied by the hinge on the rod along the rod is :-

A rod of length $\ell$ and mass m is hinged at A. It is released from horizontal position from rest. When it comes to position AB, and makes and angle of $30^{\circ}$ with vertical A ladder of length L is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment, the speed of the end in contact with the horizontal floor is v and the ladder makes an angle $\alpha = 30^{\circ}$ with the horizontal. Then the speed of the ladder's center must be :-

A small sphere of mass 1 kg is rolling without slipping on a rough stationary base with linear speed $v = \sqrt{\frac{200}{7}}$ m/s. It leaves the inclined plane at point C. Find its linear speed at point C:-

A small sphere of mass 1 kg is rolling without slipping on a rough stationary base with linear speed $v = \sqrt{\frac{200}{7}}$ m/s. It leaves the inclined plane at point C. Find ratio of rotational and translational kinetic energy of the sphere when it strikes the ground after leaving from point C:-

In the treatment of moments of inertia, introductory textbooks often present two theorems, generally called the parallel axis theorem and the perpendicular axis theorem. There is another theorem of this same genre, which is not usually included, but which is interesting and useful. It is $$ I _ {x} + I _ {y} + I _ {z} = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} $$ Here, $I_{x}$ , $I_{y}$ and $I_{z}$ are the moments of inertia about three mutually perpendicular intersecting axes, $m_{1}$ is the mass of the $i^{th}$ particle and $r_{i}$ is the distance from the intersection. The proof is simple: Taking the three axes as coordinate axes, have : $$ \begin{array}{r l} & I _ {x} + I _ {y} + I _ {z} \\ & = \sum_ {i} m _ {i} (y _ {i} ^ {2} + z _ {i} ^ {2}) + \sum_ {i} m _ {i} (z _ {i} ^ {2} + x _ {i} ^ {2}) + \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2}) \\ & = 2 \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2} + z _ {i} ^ {2}) = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} \end{array} $$ One important application is the calculation of the moment of inertia $I_{d}$ of a uniform thin-walled spherical shell, of mass M and radius R, about a diameter. Taking the centre as the origin of coordinates, have $I_{x} = I_{y} = I_{z} = I_{d}$ , and $r_{i} = R$ . The theorem gives $3I_{d} = 2\Sigma_{i} m_{i}R^{2} = 2(\Sigma_{i}m_{i})R^{2} = 2MR^{2}$ , where $I_{d} = 2MR^{2/3}$ . Consider a solid cube of mass m and side L. What will be the value of $\sum m_{i}r_{i}^{2}$ for this body when the point of intersection of axes is the centre of the cube:

In the treatment of moments of inertia, introductory textbooks often present two theorems, generally called the parallel axis theorem and the perpendicular axis theorem. There is another theorem of this same genre, which is not usually included, but which is interesting and useful. It is $$ I _ {x} + I _ {y} + I _ {z} = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} $$ Here, $I_{x}$ , $I_{y}$ and $I_{z}$ are the moments of inertia about three mutually perpendicular intersecting axes, $m_{1}$ is the mass of the $i^{th}$ particle and $r_{i}$ is the distance from the intersection. The proof is simple: Taking the three axes as coordinate axes, have : $$ \begin{array}{r l} & I _ {x} + I _ {y} + I _ {z} \\ & = \sum_ {i} m _ {i} (y _ {i} ^ {2} + z _ {i} ^ {2}) + \sum_ {i} m _ {i} (z _ {i} ^ {2} + x _ {i} ^ {2}) + \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2}) \\ & = 2 \sum_ {i} m _ {i} (x _ {i} ^ {2} + y _ {i} ^ {2} + z _ {i} ^ {2}) = 2 \sum_ {i} m _ {i} r _ {i} ^ {2} \end{array} $$ One important application is the calculation of the moment of inertia $I_{d}$ of a uniform thin-walled spherical shell, of mass M and radius R, about a diameter. Taking the centre as the origin of coordinates, have $I_{x} = I_{y} = I_{z} = I_{d}$ , and $r_{i} = R$ . The theorem gives $3I_{d} = 2\Sigma_{i} m_{i}R^{2} = 2(\Sigma_{i}m_{i})R^{2} = 2MR^{2}$ , where $I_{d} = 2MR^{2/3}$ . Find the moment of inertia of ring of mass m and radius R about an axis passing through its centre and making an angle $45^{\circ}$ with its plane :

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the z-axis is:

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the x-axis is :-

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about its base parallel to the x-axis is :-

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is $90^{\circ}$ . The apex lies at the origin and the base is parallel to X-axis The moment of inertia of the plate about the y-axis is:- $\tau = I\alpha$ Torque

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. The direction of friction on the tyres is

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. What is the maximum value of $\mu$ so that both tyres remain in contact with the ground?

A simplified model of a bicycle of mass M has two tyres that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is w, and the center of mass of the bicycle is located midway between the tyres and at height h above the ground. Air resistance may be ignored. Case - 1 (Q. 117): Assume that cycle is moving rightwards & has acceleration a. Both tyres roll purely. Case - 2 (Q. 118 and 119): The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude a. Assume that the coefficient of sliding friction between each tyre and the ground is $\mu$ , and that both tyres are skidding (sliding without rotating). Express your answers in terms of w, h, M and g. What is the maximum value of a so that both tyres remain in contact with the ground?

A thread is wrapped around a cylindrical spool of radius 1.5 cm whose central axis is fixed on a support. The thread is pulled with a constant acceleration. It is seen that it takes 2 sec to pull 6.28 cm of thread starting from rest. What is the angular velocity of the spool at 1.5 sec?

A thread is wrapped around a cylindrical spool of radius 1.5 cm whose central axis is fixed on a support. The thread is pulled with a constant acceleration. It is seen that it takes 2 sec to pull 6.28 cm of thread starting from rest. If the thread was pulled with a constant speed so that it takes 2 sec to pull 6.28 cm of thread. The angular velocity of spool at 1.5 sec is:

When an athlete takes off from the ground in a long jump, the force on the launching foot gives her an angular momentum in clockwise direction around the horizontal axis. This rotation would not allow her to land properly. In the landing, the legs should be together and extended forward at an angle so that heels marks the sand at greatest distance. Once in air the angular momentum cannot change. However, the jumper can shift most of the angular momentum to the arms by rotating them clockwise as shown in the figure. Then the body comes to proper orientation for landing. Assume that in 0.7 sec one arm covers 1/2 a revolution and another arm covers 1 revolution. Each arm is a thin rod of mass 4 kg and length 0.6 m rotating at one end. Assume that this 0.7 sec. is time of flight. What is the angular momentum of both arms in athlete's frame (in SI units)

When an athlete takes off from the ground in a long jump, the force on the launching foot gives her an angular momentum in clockwise direction around the horizontal axis. This rotation would not allow her to land properly. In the landing, the legs should be together and extended forward at an angle so that heels marks the sand at greatest distance. Once in air the angular momentum cannot change. However, the jumper can shift most of the angular momentum to the arms by rotating them clockwise as shown in the figure. Then the body comes to proper orientation for landing. Assume that in 0.7 sec one arm covers 1/2 a revolution and another arm covers 1 revolution. Each arm is a thin rod of mass 4 kg and length 0.6 m rotating at one end. Assume that this 0.7 sec. is time of flight. If rest of the athlete's body is assumed to be a rigid body having a moment of inertia of $18 \, kg m^{2}$ about its centre of mass, what is its change in angular velocity due to this rotation of arms?

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder of mass M and radius R is pulled horizontally by a force F. The friction force can be given by which of the following diagrams :-

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder is pulled horizontally by a force F acting at a point below the centre of mass of the cylinder, as shown in figure. The friction force can be given by which of the following diagrams :-

In the following problems, indicate the correct direction of friction force acting on the cylinder, which is pulled on a rough surface by a constant force F. A cylinder is pulled horizontally by a force F acting at a point above the centre of mass of the cylinder, as shown in figure. The friction force can be given by which of the following diagrams

A disc of mass m and radius R is placed over a plank of same mass m. There is sufficient friction between disc and plank to prevent slipping. A force F is applied at the centre of the disc. Acceleration of the plank is :-

A disc of mass m and radius R is placed over a plank of same mass m. There is sufficient friction between disc and plank to prevent slipping. A force F is applied at the centre of the disc. Force of friction between the disc and the plank is :-

A uniform rod of mass 4m and length L lies on a smooth horizontal table. A particle of mass m moving on the table with speed $v_{0}$ strikes the rod perpendicularly at an end and stops. Velocity of centre of mass of the rod after collision is