How Do You Calculate Dynamics of a Mass Spring System?

[deliliah]

Homework Statement

a mass spring system vibrates with an amplitude of 4.5 cm. if the spring has a constant od 250 n/m and the mass is 400 kg determine the
a) total mechanical energy
b) the maximum speed of the mass
c) the maximum acceleration
d) the speed of the mass when the displacement is 2.0 cm

Homework Equations

e= 1/2mv^2 + 1/2 kx^2
e= 1/2 ka^2 = 1/2 mv^2(max)
e=1/2 mv^2 + 1/2 kx^2
e= 1/2 mv^2 + mgh

The Attempt at a Solution

1/2 (400)(0)^2 + 1/2 (250)(4.5)^2= 2531.25?

 

[Dick]

If the amplitude is 4.5 cm then when x=4.5 cm, then v=0. Can you start with that?

 

[deliliah]

so would i just use the 1/2 mv^2 + 1/2 kx^2 for the total mechanical energy? and just plug in 4.5 for x and 0 for v?

 

[collinsmark]

1/2 (400)(0)^2 + 1/2 (250)(4.5)^2= 2531.25?

so would i just use the 1/2 mv^2 + 1/2 kx^2 for the total mechanical energy? and just plug in 4.5 for x and 0 for v?

Yes, that will work. But make sure you convert 4.5 cm to meters, if you want your final answer in units of Joules.

 

[rwisz]

a) To determine the total mechanical energy of the mass spring system, we can use the equation e= 1/2 mv^2 + 1/2 kx^2. Plugging in the given values, we get e= 1/2(400)(0)^2 + 1/2(250)(4.5)^2 = 2531.25 J. Therefore, the total mechanical energy of the system is 2531.25 J.

b) The maximum speed of the mass can be determined using the equation e= 1/2 ka^2 = 1/2 mv^2(max). We already know the value of e from the previous calculation, so we can rearrange the equation to solve for v(max). Plugging in the values, we get v(max)= √(2e/m) = √(2(2531.25)/400) = 6.31 m/s. Therefore, the maximum speed of the mass is 6.31 m/s.

c) The maximum acceleration of the mass can be determined using the equation e= 1/2 mv^2 + 1/2 kx^2. We can rearrange the equation to solve for a(max). Plugging in the values, we get a(max)= √(2e/k) = √(2(2531.25)/250) = 8.04 m/s^2. Therefore, the maximum acceleration of the mass is 8.04 m/s^2.

d) To determine the speed of the mass when the displacement is 2.0 cm, we can use the equation e= 1/2 mv^2 + 1/2 kx^2. We already know the value of e from the previous calculation, so we can rearrange the equation to solve for v. Plugging in the values, we get v= √((2e - kx^2)/m) = √((2(2531.25) - (250)(0.02)^2)/400) = 3.17 m/s. Therefore, the speed of the mass when the displacement is 2.0 cm is 3.17 m/s.