Finding a max angular acceleration

[jhayes25]

Given is a 6' long 40 lb bar hung horizontally by a pin O at its end. It is released from its horizontal position and hits a stopper after reaching a 30 degree angle with the horizontal position at which it started. I am asked to find the value x for which the angular velocity is a maximum and also the corresponding angular acceleration alpha


Sum of the torques about O=I(alpha)
a(t)=r*alpha


I honestly do not know where to start with this one. do I start by summing moments about O?
Thanks in advance for any help.

 

[tiny-tim]

Given is a 6' long 40 lb bar hung horizontally by a pin O at its end. It is released from its horizontal position and hits a stopper after reaching a 30 degree angle with the horizontal position at which it started. I am asked to find the value x for which the angular velocity is a maximum and also the corresponding angular acceleration alpha

Hi jhayes25!

(what's x? )

Hint: use the moment of inertia of the bar about its end, and net torque = rate of change of angular momentum.

 

[thomate1]

As a scientist, the first step in solving this problem would be to gather all of the necessary information, including the length and weight of the bar, the angle it reaches, and the point of rotation (pin O). From there, we can use the equation for torque (T = rFsinθ) to calculate the torque at point O and determine the angular acceleration.

Next, we can use the equation for angular velocity (ω = ω0 + αt) to find the maximum angular velocity (ω) at the point of impact with the stopper. We can also use the equation for angular displacement (θ = θ0 + ω0t + 1/2αt^2) to solve for the angle (θ) at which the bar hits the stopper.

To find the value of x for which the angular velocity is a maximum, we can use the equation for kinetic energy (KE = 1/2Iω^2) to calculate the kinetic energy at the point of impact. Then, we can use the equation for work (W = Fd) to calculate the work done by the stopper on the bar. Setting these two values equal to each other and solving for x will give us the value for which the angular velocity is a maximum.

Similarly, we can use the equation for power (P = Tω) to calculate the power at the point of impact. Setting this equal to the work done by the stopper and solving for x will give us the same value for x as before.

To find the corresponding angular acceleration (α), we can use the equation α = (ω-ω0)/t, where t is the time it takes for the bar to reach the stopper. This can be calculated using the equation for angular displacement mentioned earlier.

In summary, to solve for the maximum angular velocity and corresponding angular acceleration, we will need to use equations for torque, angular velocity, angular displacement, kinetic energy, work, and power. By carefully plugging in the given values and solving for x and α, we can determine the maximum angular velocity and corresponding angular acceleration of the bar.