What is the distance the child flies after the rope is cut?

[Daniel95]

Homework Statement

A child is sitting on a wooden stick, hanging from a 3 m long blue rope,
attached to the top of a 4 m tall metal pole. The child is performing uniform
rotation around the pole, in such a way that the angle between pole and rope
is . The horizontal distance between child and pole is called r.

The centripetal acceleration resulting
from the combined forces of rope and gravity is given by
ac = g tan
where g = 9.80 m/s2 is the gravitational acceleration.

If the period of rotation is T = 3 s, what is:

a) What is the angle ? The speed of rotation? r ? The angular speed? The
centripetal acceleration?​

Now the nasty big brother taunts the child into going faster and faster, and
keeps pushing him until his uniform motion has an angle with vertical of
= 60 degrees.

b) How fast is he now going, and what is the period of rotation? How high
up in the air is the child now zooming around?​

As the helpless child is spinning overhead, the nasty big brother ducks underneath,
reaches up and cuts the rope with a hedge-trimmer.

c) How far does the child fly off before hitting the ground? You may assume
that the child has the initial height and speed computed in b).​

2. Relevant info
So I have done a) and b) and I need some help with the c).

I have found out (in a)) that the angle is 41.9 degrees, r =2m, angular speed=2,09m/s, centripetal acceleration=8.79m/s^2, and the speed of rotation=4.19m/s
In b) I found that the speed is 6.65m/s, the period og rotation=2.46seconds and that the child is 2.5m up in the air.

How can I solve the c). I had an idea, but I've found that it was wrong.

 

[nasu]

So what was your idea? How do you know is wrong?

 

[Simon Bridge]

Welcome to PF;

How can I solve the c). I had an idea, but I've found that it was wrong.

... what was the idea?
Please show your working and not just the answers - seeing this helps us to understand how you think so we can answer your questions properly.

For (c) - consider: what normally happens to an object in circular motion when it is released?
Do you know any physics involving objects falling under gravity?

 

[Daniel95]

Please show your working and not just the answers - seeing this helps us to understand how you think so we can answer your questions properly.

Okey, so I have uploaded two pictures which shows how I have done (a) and (b).

And for (c) I first thought about using x(t)= xi + vit + 1/2 at^2 but that does not include the angle so it feels wrong..

 

[Simon Bridge]

And for (c) I first thought about using x(t)= xi + vit + 1/2 at^2 but that does not include the angle so it feels wrong..

What normally happens to an object in circular motion when it is released? It flys off in a straight line right: but which straight line? How does it move wrt the initial circle?

What were you planning to use for vi? or a or t?
Do you know any physics involving objects falling under gravity?

This is not a problem where you can just look up an equation and hope for the best, you have to understand the physics.

 

[Daniel95]

Okey, the child will clearly not fall in a straight line on to the ground. It has a velocity whitch is a tangent of the horizontal circle.

What were you planning to use for vi?

For vi I was planning to use 6,65m/s that I found in (b). And I don't know about t.

So now I am maybe thinking I can use the range equation R = vi^2 * sin(2*q) / g
If that does not work, than I don't have any idea...

 

[Orodruin]

So ask yourself the following questions:
What forces are acting on the kid after the rope has been cut?
How high above the ground is the kid at the time the rope is cut?
How long will it take the kid to hit the ground?

 

[Daniel95]

Well, aside from the air resistance, there is no other forces than gravitation acting on the child after the rope has been cut. And at that point the child is 2.5m up in the air. But still I am not sure how to find how long it will take the child to hit the ground or the distance it will fly before hitting the ground.

I feel a little ashamed, it's my first month with physics.

 

[Orodruin]

So let us ignore air resistance for the time being.

Based on that you are given this problem, I believe you will already have covered basic parabolic motion under the influence of a gravitational field, yes? What is the initial velocity in the vertical direction and how does this change with time?

 

[Daniel95]

Okey, I think I've sorted it ou based on your last question. Does this look right to you?

 

[Orodruin]

Your approach looks reasonable, I have not checked the actual numbers.

 

[Daniel95]

Wow, ok! Thank you very much, both Orodruin and Simon. Now I can sleep:)

 

[Simon Bridge]

Well done :)