A Classical Approach of Newtonian Mechanics

11 OSCILLATORY MOTION 11.6 Uniform circular motion

m 3

Let f 1 = k 1 x 1 and f 2 = k 2 x 2 be the magnitudes of the forces exerted by the

first and second springs, respectively. Since the springs (presumably) possess

negligible inertia, they must exert equal and opposite forces on one another. This

implies that f 1 = f 2 , or

k 1 x 1 = k 2 x 2.

Finally, if f is the magnitude of the restoring force acting on the mass, then force

balance implies that f = f 1 = f 2 , or

f = keff x = k 1 x 1.

Here, keff is the effective force constant of the two springs. The above equations

can be^ combined^ to^ give^

keff =

k 1 x 1

x 1 + x 2

k 1

=

1 + k 1 /k 2

k 1 k 2

=.

k 1 + k 2

Thus, the problem reduces to that of a block of mass m = 3 kg attached to a

spring^ of effective^ force^ constant^

keff =

k 1 k 2

k 1 + k 2

=

1200 × 400

1200 + 400

= 300 N/m.

The angular frequency of oscillation is immediately given by the standard formula

ω =

‚

., keff

Hence, the period of oscillation is

=

‚

., 300

2 π

= 10 rad./s.

T = = 0.6283 s.

ω

k 1 k 2

m