24-Sept-2014: Angular speed and vertical angle for particular rotating apparatus

The purpose of the lab conducted is to determine the relationship between the angular speed of the system and the vertical angle generated by the string.

The apparatus presented is a motor atop a stand which rotates a stick with a string attached to it with a mass at its end. As the stick rotated the string would generate a specific angel relative to the angular speed of the stick. As the speed increased the angel would become larger in value.

We needed to find an expression for the angular velocity of the system as it spun at numerous speeds. In order to achieve this as a class we created a free body diagram of the mass attached to the string agreeing that 3 forces were present; weight opposed by normal force cosθ in the vertical direction, and the centripetal acceleration opposed by normal force sinθ. 

Nsinθ = mr(w^2)

Ncosθ = mg

tan(θ) = (r/g)(w^2)

The value of r would equal the distance of the stick to its edge plus the horizontal component of the string generates while its in motion giving us:

r = d + l sin θ

w = √[(gtanθ)/(d+l sin θ)]

Although we have an expression for angular velocity we are still faced with the problem of changing variables that we can measure when the system is motion for safety and accuracy reasons. These variables include the vertical and horizontal components of the string length and the angle generated by the string. In order to find these important variables we must measure the total height of the system as well as the height of the mass from the ground at every speed. by doing this we found the expression:

θ = arccos [(height total - height mass) / l]

After finding all the expressions and values we put our data into excel

With the data we were able to construct a angular velocity expression vs angular velocity experimental, where expression in the angular velocity gained by the expression derived and the angular experimental being 2π divided the time recorded for the system to complete a revolution.

As a result we can see that the values are quiet similar as the slope of the graph is 0.94 where a slope of 1 would signify that the values are exactly the same.

To conclude it is not a surprise that the values were not exact since the exact time an object completes a revolution cannot be recorded by the human eye. Although the values were not identical it can still be comprehended that the greater the angle of the string the greater the angular velocity.