Initial Speed of Hanging Mass (HW Check)

[Masrat_A]

Homework Statement

A mass hangs from a light string of length 45cm attached at its other end to the ceiling. If the mass is given an initial speed of 5m/s, what is its speed when it reaches point A in its circular path (see Figure 1)?

https://imgur.com/a/ERdlD

Homework Equations

$E_i = E_f$
$KE_i + PE_i = KE_f + PE_f$

The Attempt at a Solution

$1/2mv_o^2 + mgd = 1/2v^2 + 0$
$v_o^2 + 2gd = v^2$
$2(10)(14) + 5^2 = v^2$
$17.46 = v$

I feel like there is something I'm missing, but I can't quite catch what it is...

 

[kuruman]

You are missing something very big. If the initial speed is 5 m/s, can the final speed at A be more than three times that? Check your energy conservation equation.

 

[Masrat_A]

I've thought about it, but I still can't see what went wrong... could you please give a bit more detail?

I've noticed I've made an error in my calculation:

$2(10)(14) + 5^2 = v^2$
$17.46 = v$

But the result is still much higher than initial speed:

$2(10)(45) + 5^2 = v^2$
$30.41 = v$

Should I make it so that final speed would equal initial? Point A having the same velocity as starting point does make sense to me in some capacity.

$2(10)(45) = v^2 + 5^2$
$30 = v + 25$
$5 = v$

Update: Point A would have the same potential energy as starting. Therefore:

$1/2mv_o^2 + mgd = 1/2v^2 + mgd$
$v_o^2 + 2gd = v^2 + 2gd$
$v_o^2 = v^2$
$25 = v^2$
$5 = v$

 

[kuruman]

Point A would have the same potential energy as starting.

That would be the case if point A were at the same height as the starting point. Is that true?

Perhaps you should do this piecewise. Fill in the blanks first

1. KE_i = _____

2. PE_i = _____

3. KE_f = _____

4. PE_f = _____

and then write the conservation of energy equation. You might see what you are doing wrong more clearly.

 

[Masrat_A]

That would be the case if point A were at the same height as the starting point. Is that true?

Perhaps you should do this piecewise. Fill in the blanks first

1. KE_i = _____

2. PE_i = _____

3. KE_f = _____

4. PE_f = _____

and then write the conservation of energy equation. You might see what you are doing wrong more clearly.

No, that is not right. I'm thinking the height of starting point would be rather -45cm, and at point A, we have 0.

1. KE_i = $1/2mv_o^2$ = 25
2. PE_i = $mgd$ = -875
3. KE_f = $1/2v^2$ = ?
4. PE_f = $mgd$ = 0

 

[kuruman]

More points to consider.

1. The statement of the problem does not give value for the mass. What did you use for it to get your numbers? It turns out you don't need the mass to answer this question, but you cannot assume a number and use it because it might lead you to erroneous results.
2. You are mixing units, cm for the radius and m/s for the speed.
3. What happened to the mass in your expression for KE_f?
4. Finally, I strongly recommend that you first solve the problem algebraically to get an expression for the final speed in terms of g, d and v₀ and then substitute the numbers. It is a good habit that will serve you well in the future.

 

[Masrat_A]

I see; here is another attempt.

1. KE_i = $1/2mv_o^2$ = 25
2. PE_i = $mgd$ = -9
3. KE_f = $1/2mv^2$ = ?
4. PE_f = $mgd$ = 0

$1/2mv_o^2$ + $mgd$ = $1/2mv^2$ + 0

The masses cancel each other, thus...

$v_o^2$ + $2gd$ = $v^2$

Converting cm to m, we have:

$5^2$ + $2(10)(-0.45)$ = $v^2$
$16$ = $v^2$
$v = 4$

 

[kuruman]

That looks correct.