Are you saying the given answer to "the question" is ##a_y=\frac 12L\cos(\theta)\dot\omega##? If so, the question must be "find the relationship between the linear and angular accelerations", right?Sam Jelly said:
Yes, the question is to find the linear acceleration in term of the angular acceleration. But my answer seems to be different.haruspex said:
Wait… is the question only asking for the initial acceleration when the rod is released from rest? At that time, ##\omega=0##, so the answers are the same.Sam Jelly said:
It’s the acceleration at any point in time.haruspex said:
Ok… but why then does it say theta is the initial angle, not the angle at an arbitrary time?Sam Jelly said:
Oh yes I forgot the -sin theta.haruspex said:
Ok.Sam Jelly said:
When a rod begins to slide or fall on a frictionless horizontal plane, it experiences no frictional force acting against its motion. As a result, the rod will accelerate due to gravity if it is falling, or it will maintain its velocity if it is sliding horizontally, as there are no forces to decelerate it.
When the rod is sliding on a frictionless horizontal plane, the only force acting on it is its weight, which acts downward. There are no horizontal forces since the plane is frictionless. In contrast, if the rod is falling, it is also subject to gravitational force, but its orientation may cause it to rotate as it falls, depending on its pivot point and initial conditions.
The center of mass of the rod will follow a straight-line path while sliding on the frictionless plane, moving at a constant velocity. If the rod is falling, the center of mass will accelerate downward due to gravity, following a parabolic trajectory if there is an initial horizontal velocity component.
The length of the rod affects its moment of inertia, which influences how it rotates if it is falling. However, on a frictionless horizontal plane, the length does not affect the translational motion since the absence of friction means that all parts of the rod will slide uniformly without any angular acceleration unless acted upon by an external torque.
No, a rod cannot maintain static equilibrium while sliding or falling on a frictionless surface. If it is sliding, it is in motion and cannot be in equilibrium. If it is falling, it is accelerating due to gravity, and thus also cannot be in equilibrium. Equilibrium requires a net force of zero, which is not the case in either scenario.