Two Circular Motion Questions. Pendulums

[navm1]

Homework Statement

Got my first physics assignment and I've been able to work out (I think) all of them except these two.

1. A conical pendulum has length 1.5m and rotates at 4ms^-1. What is its angle to the vertical?

2. A mass moves in a vertical circle attached to a fixed point by a string of length 2m. How fast must it be moving at the lowest point if it completes circles? What would be the speed necessary at the lowest point if the mass were attached by a light rod instead of a string?

Homework Equations

F=ma
F=mv^2/r
v=rω
ω=θ/r

The Attempt at a Solution

1. I've drawn a diagram and understand that i have gravity acting downwards and tension up the string. I think i could find θ by using tanθ=v^2/rg but I am not sure of how to work the radius out if i only have the length of the string, the velocity, and gravity.

2.
I'm only given the length and gravity on this question. I think i might have to work out the tension but I am finding it difficult without knowing a mass of the object.

I understand my attempts didn't go too far. Tension, work, and energy were omitted from my course for some reason so I've got some catching up to do with it. Thanks

 

[tiny-tim]

hi navm1! welcome to pf!

(try using the X² button just above the Reply box )

1. I've drawn a diagram and understand that i have gravity acting downwards and tension up the string. I think i could find θ by using tanθ=v²/rg but I am not sure of how to work the radius out if i only have the length of the string, the velocity, and gravity.

either eliminate T by doing F = ma for both the x and y directions,

or ignore T by choosing … which direction?

2.
I'm only given the length and gravity on this question. I think i might have to work out the tension but I am finding it difficult without knowing a mass of the object.

call the mass "m" … it'll cancel out in the end

 

[navm1]

Thanks
I'm still really struggling with these. Any pointers on where to start? I was told the first one will include a quadratic equation somewhere

 

[ehild]

1. A conical pendulum has length 1.5m and rotates at 4ms^-1. What is its angle to the vertical?

Homework Equations

F=ma
F=mv^2/r
v=rω
ω=θ/r

The Attempt at a Solution

1. I've drawn a diagram and understand that i have gravity acting downwards and tension up the string. I think i could find θ by using tanθ=v^2/rg but I am not sure of how to work the radius out if i only have the length of the string, the velocity, and gravity.

Look at your diagram (or the yellow triangle on the attached picture): how is the radius r related to the length of the string and the angle θ?

ehild

 

[tiny-tim]

hi navm1!

(just got up :zzz:)

I'm still really struggling with these. Any pointers on where to start?

start with F = ma for both the x and y directions (separately) …

show us what you get ​

 

[navm1]

hi navm1!

(just got up :zzz:)


start with F = ma for both the x and y directions (separately) …

show us what you get ​

me too. I'm sick so its a perfect day to stay in and do physics all day.

so the force of gravity is (m)(-9.8) only in the y direction which means the y component of the tension must be (m)(9.8). the only two forces acting are gravity and tension so the x component of the tension must be providing the centripetal force on its own. so i drew a diagram for tension and ended up with (9.8)tanθ=v²/r and that's as far as i made it so far without knowing angle θ

 

[haruspex]

(9.8)tanθ=v²/r

Right so far. You know v and the length of the string. What other equation can you write relating these with θ and r?

 

[tiny-tim]

… (9.8)tanθ=v²/r and that's as far as i made it so far without knowing angle θ

you know L = 1.5, so what's the relation between L r and θ ?

 

[navm1]

i don't know if I am going down the route you guys have suggested but i ended up changing my equation from

(g)(tanθ) = v²/r

to

sinθ/cosθ = v²/(l*sinθ)(g)

then i think i could square the sinθ and start to build a quadratic equation from there?

if i multiplied both sides by sinθ would i end up with

sin²θ/cosθ = v²/lg ?

edit: or could i replace sin²θ with 1-cos²θ

 

[navm1]

so if x = cosθ
then my equation is x²+(v²/lg)*x-1=0. v²/lg

i came out with x=0.6 so cos^-1 would be 53.1°
which would make the vertical angle 36.9°

am i on the right track here? thanks

 

[tiny-tim]

(g)(tanθ) = v²/r

sinθ/cosθ = v²/(l*sinθ)(g)

sin²θ/cosθ = v²/lg ?

edit: or could i replace sin²θ with 1-cos²θ

so if x = cosθ
then my equation is x²+(v²/lg)*x-1=0. v²/lg

yes!

how are you getting on with question 2?

2. A mass moves in a vertical circle attached to a fixed point by a string of length 2m. How fast must it be moving at the lowest point if it completes circles? What would be the speed necessary at the lowest point if the mass were attached by a light rod instead of a string?

 

[navm1]

yes!

how are you getting on with question 2?

so did I get the correct angle for the vertical angle? thanks a lot for the help

Gonna start question 2 as soon as i know I've got question 1 right then I'll let you know how i get on

 

[tiny-tim]

so did I get the correct angle for the vertical angle? thanks a lot for the help

looks right to me

 

[navm1]

so for the second question i have the equations

v=√v_i² - 2gl(1-cosθ) and T(tension)= (mv_i² / l ) - 2mg+3mg*cosθ

are these the right equations to be using given that i only have the length of the string and nothing else?

 

[tiny-tim]

so for the second question i have the equations

v=√v_i² - 2gl(1-cosθ) and T(tension)= (mv_i² / l ) - 2mg+3mg*cosθ

(= mv²/l + mg*cosθ)

yes

 

[navm1]

thanks. I am havin a little trouble working out how i'd find a velocity if i don't have an initial or final velocity. if i want to find out the speed it needs to be at 180 degrees id have theta as well as l and g but no velocity

edit: i had 0=u^2-2gl(1-cosphi) then switched final velocity to make initial velocity on the outside to try and find the velocity needed for it to reach the top where its velocity would be zero but i got a math error

 

[tiny-tim]

2. A mass moves in a vertical circle attached to a fixed point by a string of length 2m. How fast must it be moving at the lowest point if it completes circles? What would be the speed necessary at the lowest point if the mass were attached by a light rod instead of a string?

thanks. I am havin a little trouble working out how i'd find a velocity if i don't have an initial or final velocity …

what is the lowest possible value of the tension if the mass moves in a complete circle?

 

[navm1]

does it have to be at least 9.8N?

edit: i think if it reaches 0 near the top then the string will slack and it will start free falling affected only by mg

 

[haruspex]

does it have to be at least 9.8N?

edit: i think if it reaches 0 near the top then the string will slack and it will start free falling affected only by mg

Those are two different answers. Does the tension need to be at least mg or at least zero (+ a tiny bit)?

 

[tiny-tim]

(just got up :zzz:)

edit: i think if it reaches 0 near the top then the string will slack and it will start free falling affected only by mg

that's much closer!

but the string will be slack only if the tension is exactly zero …

(it can't be less than zero, and) if it's greater than zero, then the string must be its full length: which means that the mass stays on the circle!

(this is very similar to the fact that something will leave a surface only if the normal force is exactly zero)