Maximizing Projectile Motion: Finding the Optimal String Cut Location

[oscar_lai_hk]

A mass is attached to one end of the massless string, the other and of which is attached to a foxed support. The mass swings around in a vertical circle as shown in Fig 5.36. Assuming that the mass has the minimum speed necessary at the top of the circle to keep the string from going slack, at what location should you cut the string so that the resulting projectile motion of the mass has its maximum height located directly above the center of the circle.

 

[tiny-tim]

A mass is attached to one end of the massless string, the other and of which is attached to a foxed support. The mass swings around in a vertical circle as shown in Fig 5.36. Assuming that the mass has the minimum speed necessary at the top of the circle to keep the string from going slack, at what location should you cut the string so that the resulting projectile motion of the mass has its maximum height located directly above the center of the circle.

Hi oscar_lai_hk!

Show us what you've tried, and where you're stuck, and then we'll know how to help.

 

[oscar_lai_hk]

Hi oscar_lai_hk!

Show us what you've tried, and where you're stuck, and then we'll know how to help.

I can write the equation towards the center : T + mg cos0 = mv^2/R
and I consider the energy at the moment of escape and the highest position
I get the equation (0.5)mv^2 = (0.5)m(vcos0)^2 + mgy
then I try to diff. the above equation of y respect to the angle 0
it is right?

 

[tiny-tim]

I can write the equation towards the center : T + mg cos0 = mv^2/R
and I consider the energy at the moment of escape and the highest position
I get the equation (0.5)mv^2 = (0.5)m(vcos0)^2 + mgy
then I try to diff. the above equation of y respect to the angle 0
it is right?

Hi oscar_lai_hk!

I'm really not following that.

Start by putting T = 0 at the top of the loop to find out what v_top is.

Then find out what v_θ is, at any angle θ.

The split v_θ into horizontal and vertical components …