a uniform bar of mass is supported by support pivoted at top about which bar can swing like simple pendulum. If force F is applied perpendicular to lower end of the bar.How big F must be in order to held the bar in equilibrium.
That will depend upon the angle A that the bar makes with the vertial axis.
The torque due to the weight of the bar, measuered about the pivot, but equal the applied torwue, F*L
F*L = M*g*sin A*(L/2)
(L/2 is the distance of the center of m, from the pivot).
Solve for F. L cancels out
F = (M g sinA)/2
To determine the force required to hold the bar in equilibrium, we need to consider the conditions for rotational equilibrium.
In this scenario, the bar is being supported at the top, allowing it to swing like a simple pendulum. Let's assume that the length of the bar is "L" and the mass of the bar is "m". The force, "F", is applied perpendicular to the lower end of the bar.
To keep the bar in equilibrium, the net torque acting on it must be zero. Torque, denoted by "τ" is given by the equation:
τ = r * F * sin(θ),
where "r" is the distance from the pivot point to the point where the force is applied, "F" is the force applied, and "θ" is the angle between the force and the line connecting the pivot to the point of application.
In this case, since the force is acting perpendicular to the lower end of the bar, the angle "θ" is 90 degrees, and sin(θ) = 1.
Now, let's consider the torque acting due to the weight of the bar. The weight vector acts at the center of mass, which is at a distance of L/2 from the pivot. The weight is given by the equation:
W = m * g,
where "g" is the acceleration due to gravity.
The torque due to the weight is:
τ_weight = (L/2) * (m * g) * sin(90) = (L/2) * (m * g).
To keep the bar in equilibrium, the applied force must generate an equal and opposite torque to counterbalance the torque due to the weight.
Thus, the required force "F" can be calculated using the equation:
F = (τ_weight) / (r * sin(θ)),
F = [(L/2) * (m * g)] / [r * sin(θ)].
Please note that to get an exact value for "F," you need to provide the specific values for "m," "L," "g," "r," and the angle "θ."