A 0.50-kg ball that is tied to the end of a 1.4-m light cord...

A 0.50-kg ball that is tied to the end of a 1.4-m light cord is revolved in a horizontal plane, with the cord making a ? = 30° angle with the vertical. (a) Determine the ball's speed. (b) If, instead, the ball is revolved so that its speed is 3.7 m/s, what angle does the cord make with the vertical? (c) If the cord can withstand a maximum tension of 9.6 N, what is the highest speed at which the ball can move?

Transcript

00:01 All right, so we have a ball being whirled around in a circle like this, and this circle, or this ball is suspended from a cable that has a length of 1 .4 meters.

00:13 And it's initially at a 30 degree angle.

00:17 And so we want to know what is the speed of the ball, given that it has a 30 degree angle here.

00:23 Also, the ball has half a kilogram mass, which we'll get to that later.

00:26 So we draw a free body diagram.

00:28 The tension in the cable is going up like this the weight is going down like this and the centripetal force is going inwards mb squared over r and so we can also write the tension is like being under the triangle or underneath the ball there's nothing wrong with that because then we can see the relationship between all these components of this vector and so we can see that the tangent of this angle is equal to, and then if we look at it, it's v squared over r times g.

01:07 And r note, the radius of the circle is going to be l times the sine of theta.

01:14 L is like the length of the cable.

01:16 So for the case of 30 degrees, r is the sign of 30 times 1 .4, or 0 .7 meters.

01:24 And so if we solve for this, v we see is the square root of r times g times the tangent of theta.

01:32 So this is 0 .7 meters times 9 .8 meters per second squared times the tangent of 30, and we take the square root of all that, should get something like 1 .99 meters per second.

01:51 Next up, we want to know if we increase the speed, so this is question b, if we increase the speed to 3 .7 meters a second, what happens to the angle? what does the angle become? well, presumably the height that this is being rotated from doesn't change.

02:08 It's just we change the speed.

02:09 And so the relationship between the angle and the height is going to save us here, because you probably got stuck for the same reason many people get stuck on this problem, is that you realize the radius depends on the angle as well.

02:20 So when you're calculating, if you use the expression, the tangent of the angle is equal to v squared over r times g.

02:29 Well, r depends on the angle as well.

02:31 But we can also write r...

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