The motion of the uniform rod A B is guided by small wheels...

The motion of the uniform rod $A B$ is guided by small wheels of negligible mass that roll on the surface shown. If the rod is released from rest when $\theta=0,$ determine the velocities of $A$ and $B$ when $\theta=30^{\circ} .$

Key Concepts

Conservation of Energy in Dynamics

Conservation of energy in dynamics involves equating the loss of potential energy to the gain in kinetic energy (both translational and rotational) in a system. This principle is especially useful in problems where direct analysis via forces is complicated, as it allows one to determine velocities and other kinematic quantities by considering energy transformations along the motion path.

Kinematic Constraints

Kinematic constraints are the geometry-based restrictions that connect the motions of different parts of a system. When multiple bodies or points are connected through guides or rolling elements, these constraints allow the derivation of relationships between their velocities and accelerations, facilitating the analysis of complex coupled motions in mechanisms.

Rigid Body Kinematics

Rigid body kinematics studies the motion of a solid body where all points remain in fixed relative positions. It relates the translational motion (movement of the center of mass) and the rotational motion (angular velocity about an axis) to determine the velocity of any point on the body using its angular velocity and the geometry of the body.

Rolling Without Slipping

The rolling without slipping condition ensures that at the point of contact between a rolling object and the surface, there is no relative sliding. This creates a precise link between the linear speed of the object and its angular speed, making it critical in problems where motion is guided by surfaces, as it constrains the velocity relationship between different parts of the system.

Transcript

00:01 As shown in the figure a uniform rod a v is guided by small fields of negligible mass which are rolling on the surface as shown in the figure if rod is released from rest that is omega 1 is 0 at theta to v 0 calculate velocities of a and b for theta to be 90 degree.

00:53 Let us see here.

00:54 For position 1, theta is 0 degree.

01:11 Velocity of a is called to v is called to 0.

01:17 Angular velocity is 0.

01:20 Kinetic energy is zero.

01:30 Potential energy is zero.

01:37 Per position theta is called to 30 degree.

01:54 If we draw the diagram, this is the rod a, this is 30 degree this is 60 degree this is 60 degree this is 60 degree this will be 60 degree this is the velocity a and this is the velocity b this is center of mass of the rod and these are the 30 degree angle each so this is the geometrical diagram ac is called to b c is equal to l so for position to angle becomes 30 degree c is instantaneous center of c is instantaneous center triangle a bc is equator triangle you may see in the figure magnitude of velocity of a is called to b that can be given by omega into l velocity of centroid or center of mass will be omega l cost of 30 moment of inertia will be 1 by 12 m l square kinetic energy t two half m velocity of center of mass plus half i omega square substituting the value velocity of center of mass is omega l cost of 30 moment of intercept the rod about the central oil already we have written m l square by 12 and that is omega square so t2 you will get 5 by 12 m omega square l square now potential energy of the system minus m g l by 2 sign of 30 that is minus 1 by 4 m g l now applying conservation of energy so according to conservation of energy we can write initial kinetic energy plus initial potential energy final kinetic energy final kinetic energy final potential energy initial kinetic energy is zero potential energy final kinetic energy we have obtained 5 by 12 m l square omega square and potential energy is minus 1 by 4 m g l...

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