Having Maximum Velocity Issues

[John_b]

First post. You guys have really been helping me out I am trying to do a course distance learning and often the course just doesn't give you enough help and the stuff on here has been great. I have had a look at previous posts but just can't seem to get my head round this one.

The Problem:-

A mass of 0.3kg is suspended from a spring with stiffness 200Nm-1. The mass is displaced by 10mm from its equilibrium position and released, for the resulting vibration calculate:

ai) the frequency of vibration
aii) the Max velocity of the mass during vibration
aiii) the Max acceleration of the mass due to vibration
aiv) the mass required to produce double the max velocity calculated in (ii) using the same spring and initial deflection.


My attempt at a solution is as follows:-

ai) F= 1/2x(pi) x root K/m = 0.159 x root 666.667 = 4.11 Hz
aii) This is where my problems start I always calculate the average speed essentially using distance/time. can I calculate the Energy = mgh then use E = 0.5*mv^2??
aiii) Once I have calculated V I would use a=v-u/t.
aiv) I think I'm on the correct lines with this one using, F=1/2(pi) x root K/m then transpose so m is the subject. my answer for this one came out as 0.075kg

Any help would be much appreciated. Thanks guys.

John_B

 

[NoPoke]

spring plus mass and no damping = simple harmonic oscillator .

 

[VantagePoint72]

aii) The potential energy stored in a stretched spring (stretched by a distance $x$) is $U=\frac{1}{2}kx^2$ (the only effect gravity has is to shift the spring's equilibrium position; as long as you measure from this new equilibrium position, gravity can be ignored). So this is the formula for potential energy you should use, not $U=mgh$. Otherwise, you are correct: by equating this with the kinetic energy, you will obtain the maximum velocity of the mass. Do you understand why? Think about conservation of energy and how the amount of total energy that's stored as potential energy and the amount stored as kinetic energy varies with the mass's motion. At what point would you expect the mass to have its max. velocity? How does its kinetic energy compare with its potential energy at this point?

aiii) What you're suggesting would only give you an average acceleration. The acceleration of the bob (technical name for the mass) is different at every point. Think in terms of forces instead of going straight to acceleration (another way of saying this: think about the problem dynamically instead of kinematically). Hooke's law tells you the force acting on the mass at every point of its oscillation. Now think about it for a minute: what law do you know that relates the force acting on an object to how much it accelerates?

aiv) Try this again once you've correctly solved ii. That problem will give you a relationship between the mass of the bob and the maximum velocity; use that relationship.