Rotational Kinematics -05- Angular Velocity of a Ball Connected to a String

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A ball of mass m connected to a string, one end of which is fixed to a ceiling, rotates horizontally. If the length of the string is 1 meter and the angle of deflection of the string from the vertical is 30°, find the angular velocity of the ball.  Watch the video about ROTATIONAL KINEMATICS in full: https://dailymotion.com/playlist/x8z9ke  #rotationalkinematics #rotationalmotion

Transcript

00:00Hey folks, Monday morning is a great time to reflect on your goals, your vision, and your well-being.

00:10A ball of mass is attached to a string that is attached to a ceiling.

00:16The ball is then deflected 30 degrees about the vertical axis.

00:22If the ball rotates horizontally, what is its angular velocity?

00:30This problem seems simple, but is it really?

00:35As a starting point, let's assume the string is attached to the center of coordinates.

00:40This means the ball is rotating about the y-axis.

00:47We'll look at this system from the front.

00:52The length of the string is L, the radius of rotation is R, and the angle of deflection of the string is theta.

01:00It looks like the ball has a mass of M.

01:05An object moves because of the resultant force acting on it.

01:11From the previous series, we know that the force that causes an object to rotate is the centripetal force.

01:17It's better if we identify what forces are acting on this system.

01:21This system consists of only one body.

01:27The object with mass is subjected to a vertical downward gravitational force.

01:32The rope is tauted, and there is a tension force on the rope.

01:37This seems to be the only force acting on the ball.

01:40It turns out that none of these forces are in line with the radius of rotation.

01:47We must decompose this force into component vectors.

01:52Now it is clear that the force directed towards the center of the circle is t sine theta.

01:56According to Newton's laws of motion, t sine theta is equal to Mv squared over R.

02:07For a force in the vertical direction, the ball does not move in that direction.

02:12So, t cosine theta minus mg equals 0.

02:15We can write this equation as t cosine theta.

02:24We have two equations with three unknowns, Mv, T, and V.

02:33Simply divide the two equations to eliminate the values of T and M.

02:38Sine theta over cosine theta is equal to V squared over GR.

02:42Unfortunately, what is being asked is angular acceleration.

02:50In uniform circular motion, V is equal to R omega.

02:56One more thing.

02:58The length of the radius is L sine theta.

03:01Because we are given the length of the string, not the radius of rotation.

03:05From here, omega is equal to the square root of G over L cosine theta.

03:14Now we can substitute some values from the problem sheet.

03:17I think we should use a scientific calculator.

03:25Omega is about 3.4 radians per second.

03:29You may get different values.

03:34Happy learning everyone.

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