Question

Consider the data for a block of mass

m = 0.245 kg

given in the table below, which gives the position of a block connected to a horizontal spring at several times. Friction is negligible.

Time of measurement, t (s)	Position of block, x (m)
0	4.75
0.25	3.36
0.50	0
0.75	−3.36
1.00	−4.75
1.25	−3.36
1.50	0
1.75	3.36
2.00	4.75
2.25	3.36
2.50	0

(a) What is the mechanical energy of the block–spring system?
J

(b) Write expressions for the kinetic and potential energies as functions of time. (Use the following as necessary: t. Assume K and U are measured in joules. Do not include units in your answer.)

K	=
U	=

Answer 1

From the table, we can see that the motion is a simple harmonic motion.

The amplitude of the SHM is 4.75 m and the period of SHM is 2 s.

An SHM can be described by a cosine function. i.e

Putting the values in this equation, the equation becomes

The above equation gives us the position of the block as a function of time

The velocity of the block as a function of time can be calculated using

Now,

PART A:

the mechanical energy of the system is the sum of the kinetic and potential energies at any given time. We can do some smart work here and calculate the mechanical energy at a time when the potential energy is zero.

The potential energy is given by

Let us put this equals zero

We know that at t = 0.5 we have x = 0.

So, let us see the velocity of the system as t = 0.5.

So, the kinetic energy of the system at this position is

So, the total mechanical energy of the system is

_________________________________________________

PART B:

The kinatic energy of the system as a function of time is

And the potential energy of the system as a function of time is