Question

A toy manufacturer would like to keep the acceleration of a foam projectile down to a reasonable 15 m/sec^2. What is the mass limit the toy manufacturer could use with a spring that has a spring constant of 49 N/m and is compressed 3.5 cm?

Answer 1

@SolomonZelman

Answer 2

Let's apply Newtons' Second law.
\(\color{#000000}{ \displaystyle \sum \vec{\rm F}={\rm m}\vec{\rm a} }\)
\(\color{#000000}{ \displaystyle {\rm F_{spring}}+{\rm W}={\rm m}\vec{\rm a} }\)
\(\color{#000000}{ \displaystyle (-k\cdot x)+({\rm m}\cdot g)={\rm m}\vec{\rm a} }\)

\(\color{#000000}{ \displaystyle (-49\cdot 0.035)+({\rm m}\cdot 9.80)=15{\rm m} }\)

Answer 3

You have the Force of spring acting up on the box, and you have the weight acting down on the box, and your total (net) acceleration needs to be 15m/s^2. This is why I set it up like this.

Answer 4

Btw, in case you wondered, the weight is positive, because we want to look for acceleration down, so I changed my down to be the +direction, and up to be the -direction.

Answer 5

Wait, I came out with a negative mass xD

Answer 6

Ok. Let's see... I noticed that. So the nonnegative(lol) final result would be .32 kg or 320 grams.

Answer 7

If I get a negative mass, it might mean an error in setup.

Answer 8

just want to make sure, are you sure that these are the numbers and all units in the question are right?

Answer 9

\[15m^2 , 49\frac{ N }{ m }, 3.5cm\]

Answer 10

Those are the numbers and units in this question, to clarify(checked hard copy).

Answer 11

OK.

Answer 12

Well, let's re-check it by assigning + to up, and - to down.

Answer 13

F_spring is then positive, since it is acting up (towards the equilibrium from where it was compressed).
15m/s^2 is a negative acceleration, since it is acting down.
Weight is negative.

\(\color{#000000}{ \displaystyle (49\times 0.035)-9.8m=-15m }\)

Answer 14

But, then again I get a negative mass with same magnitude. Ahhh

Answer 15

Yep, I got the same :|

Answer 16

Let's suppose that the spring was compress, and its equilibrium is below, so towards equilibrium is downwards, although the question, I don't think it sounds like this.

Answer 17

\(\color{#000000}{ \displaystyle (-49\times 0.035)-9.8m=-15m }\)

Answer 18

Yes, then I get:
\(\color{#000000}{ \displaystyle {\rm m} =0.3298~kg }\)

Answer 19

Oh, it's compressed, not stretched, so I think it actually makes sense that the spring force is acting downward.

Answer 20

Yeah, I got the same as well. m = 0.329807
That makes sense that the compressions would indicate "downward" force.

Answer 21

Yeah. I am a legitimate source when it comes to physics, so in case, you might want to recheck...

Answer 22

Good Luck:)

Answer 23

Yeah, thanks for the help! I will check it for sure.

Answer 24

Alrighty ... have fun .Oo