Solving Mechanics Problem: Angle, Speed, Centripetal Acceleration & More

[Torunn]

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).

Homework Equations

Can someone help me with a? don't know how to begin. I have found the angular speed but that's it.

The Attempt at a Solution

 

[Orodruin]

I would like to help you but you need to show your effort so I can understand where you are having problems. You mentioned uniform motion - what do you know about uniform circular motion? (If the answer is "nothing" I recommend you start by reading about it in your course material and return if you have specific questions.)

 

[Torunn]

Have read the whole chapter about it, i know the object is moving in a circular path with constant speed. The problem is that when I try to solve these equations there always seems to be two unknowns. Have used omega=2pi/T to find the angular speed. I feel I need just a hint in the right direction to figure out this problem.

 

[Orodruin]

You say you have tried to solve "the equations". What equations are those and what is their meaning? In order to help you I need to know where you get stuck and what information you are missing.

 

[Torunn]

ac=v^2/r
T=2pi*r/v
ac=g*tan
but those two have several unknowns.
and have also tried to rearrange omega=2pi/T so that you get v=r*omega, but no matter how i rearrange them i can't find any answers.
i would normally try to find the angle first since its the first question, but know i can't find the answer.

 

[Chestermiller]

Have you drawn a free body diagram yet? If so, please tell us your force balance equation in the horizontal direction?
Chet

 

[Torunn]

we have a drawing, but in this problem we are not supposed to use Newtons laws if that is what you mean.

 

[Chestermiller]

we have a drawing, but in this problem we are not supposed to use Newtons laws if that is what you mean.

If you're not supposed to be using Newtons laws on this mechanics problem, what are you supposed to be using?

Chet

 

[Torunn]

Vectors, and xf=xi+ vi*t + 1/2at^2
and, Vf=vi+at
its the chapter "motion in two dimensions",
next chapter is the laws of motion, so then we start to use a full body diagram and Newton.

 

[Orodruin]

I agree that you do not need to consider forces any more as you have already been given the result of the free body diagram computation ($a_c = g \tan\theta$ - however, the free body diagram would be useful in reaching this conclusion).

However, you have just listed three equations and you have four unknowns. Thus, your information is not sufficient and you need to find another equation. My hint here is to think of the geometry of the problem: How can you relate the angle $\theta$ with other things that you already know (or that is a variable)?

 

[Torunn]

r= sin(theta)*3 is the one I would use to find the r or angle.

 

[Orodruin]

Yes, so you now have four equations and four unknowns. This means you can solve your system of equations. Solve for an unknown (for example ac) in one equation in terms of the other unknowns, once this is done you can insert your result into the other equations and you will have a system of equations with three equations and three unknowns (in the case of first solving for ac, the unknowns will be r, v, and theta). Repeat the process until you have found one of the variables, then start putting things back in.

 

[Torunn]

I put the two different ac equal to each other, and then i found theta by solving that one.