Moving object on a string

[victorias]

If we have a mass on a string and swing it - I am confused about at what points there would be the maximum tension , maximum velocity etc.

In one of the practice problems, they stated that max tension = mg + 1/2mv^2
But when the string is perfectly perpendicular to the ground, the point at which I though there would be the maximum tension, don't we assume that the object is still (not moving) for a brief period of time - why do we have to take KE into account?

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[aldol16]

Okay, think about it this way: conservation of energy. In this problem, you're basically converting between two forms of energy - gravitational potential energy and kinetic energy. When the mass is parallel to the ground, all its energy is in the form of gravitational potential energy - i.e. it's not moving. You also know this because at one instant, it's moving up and then at the next instant, it's moving down. So therefore, velocity went from positive to negative and you know that whenever something goes from positive to negative, it has to pass zero.

Similarly, when the object is perpendicular to the ground, all its potential energy has been converted to kinetic energy. Thus, it must be moving fastest there.

 

[victorias]

At the point when the object is perpendicular to the ground, T = mg + 1/2mv^2 but the units don't seem to match up. KE is Joules and mg is Newtons. How does energy factor into the case if we are concerned with tension (a type of force)?

How would you determine Tension when the string is horizontal, at a point with maximum potential energy?

 

[aldol16]

Uhh, well I think you've got your formula wrong. Think about it this way. What forces are acting on the object when it's perpendicular to the ground? You have its weight. You have the tension force. That's it. In this case, sum of these forces is providing the centripetal force that keeps it from flying off. So in other words: Fc = T - mg. Now, what is Fc? Well, that's just m*v^2/r. So you get m*v^2/r = T - mg, or T = mg + m*v^2/r. So I think you have some equation mixed up.

Tension when the string is horizontal can be determined using the same equation. You should have an intuitive sense of what tension should be in that case (Hint: the object is no longer moving so is there a need for a centripetal force?). In terms of equations, you're interested in only the radial component of gravity at all times, i.e. the component that counteracts the tension force. That is determined by the angle of the string, theta. Draw the diagram and you'll see that that component of gravity is m*g*cos(theta). So T = m*g*cos(theta) + m*v^2/r. That will give you tension at any point.