Pendulum Swinging Speeds: Solving for Initial Velocity and Angle

[Arman777]

Homework Statement

A pendulum of length $L=1.25m$.Its bob (which effectively has all the mass) has speed $v_0$ when the cord makes an angle $θ_0=40.0^°$ with the vertical.$(a)$ What is the speed of the bob when it is in its low (est position If $v_0=8.00m/s$ ? What is the least value that $v_0$ can have If the pendulum is to swing down and then up $(b)$ to a horizontal position,and $(c)$ to a vertical position with the chord remaining straight ? $(d)$ Do the answers to parts $(b)$ and $(c)$ increase,decrease,or remain the same if $θ_0$ is increased by a few degrees ?

Homework Equations

$-ΔU=W$
$W=Δ(KE)$

The Attempt at a Solution

[/B]
I found the answer of (a) using -
$-ΔU=W$
$W=Δ(KE)$
it came $v=8.35\frac m s$

I didnt understand (b) and part (c).so I stucked there...

Thanks

 

[haruspex]

I didnt understand (b)

Do the same calculation as you did for part a) to get an expression for the speed v at any given angle, θ, in terms of v₀, θ₀, L, g and θ.

 

[Arman777]

$\sqrt {(v_0)^2+2gL(cosθ-cosθ_0)}=v$

 

[haruspex]

$\sqrt {(v_0)^2+2gL(cosθ-cosθ_0)}=v$

Ok. So what is θ in b)?

 

[Arman777]

90 degree

 

[haruspex]

90 degree

So plug that in. What is the smallest value v₀ can have?

 

[Arman777]

Ok I found but for c what would be the angle ?

 

[haruspex]

Ok I found but for c what would be the angle ?

θ₀? Same as before.

 

[Arman777]

θ₀? Same as before.

θ ?

 

[haruspex]

θ ?

You told me in post #5.
I have the feeling that there is something you are not understanding about the question, and I can't figure out what it is.

 

[Arman777]

I found the answer of (b),plotting θ=90
For (c) what is θ=?

 

[haruspex]

I found the answer of (b),plotting θ=90
For (c) what is θ=?

I'm very sorry, my eyes kept jumping to the wrong line, so I was reading b instead of c.
For c there is a complication. For b, the speed at the horizontal position could be 0. Could the speed be zero at the top of the loop?

 

[Arman777]

I'm very sorry, my eyes kept jumping to the wrong line, so I was reading b instead of c.
For c there is a complication. For b, the speed at the horizontal position could be 0. Could the speed be zero at the top of the loop?

It cant

 

[haruspex]

It cant

Right. So what is the minimum speed at the top.

 

[Arman777]

$v=2\sqrt {gL}$

 

[haruspex]

$v=2\sqrt {gL}$

No. Consider this condition

with the cord remaining straight

What forces act on the bob at the top? What must the net force be?

 

[Arman777]

No. Consider this condition

What forces act on the bob at the top? What must the net force be?

Yeah I made a mistake I know I noticed now

 

[Arman777]

v=6.72

 

[haruspex]

v=6.72

That's not right for the speed atthe top. How did you get that? Or is that your answer for v₀?

 

[Arman777]

At the bottom total energy is 1/2mv^2 which here v is as we found ot 8.35 and at the top energy is 1/2mv^2+mg2L so I thought the should be equal

 

[haruspex]

At the bottom total energy is 1/2mv^2 which here v is as we found ot 8.35 and at the top energy is 1/2mv^2+mg2L so I thought the should be equal

For c, you are finding the initial velocity such that it will reach the top with a straight string. You do not know the KE earlier.
Forget energy for the moment and think about forces. What is the minimum speed at the top for the string to remain straight?

 

[Arman777]

T-mg=mv²/r

 

[haruspex]

T-mg=mv²/r

Right. What is the minimum value of T?

 

[Arman777]

I solved thanks :)