Question

Two masses, m1 and m2 hang over a frictionless pulley by means of a massless string. They are perfectly balanced when a flea of mass m3 is standing on m2. When the flea hops from one mass to the other, the system acquires an acceleration, a. What is the value of the flea's mass, m3 in terms of two masses, m1 and m2, and the acceleration, a? (a) what is m3 in terms of m1 and m2? (b) what is m3 in terms of m1 and a? (c) what is m3 in terms of m2 and a?

Answer 1

a) let T be the tension in the string

When the system is at rest,

m1g = T and

T =( m2+m3)g

m3 = m1-m2

Now whne the feal hops from m2 to m1 the system set into acceleration. m1 moves down and m2 moves up, let a be the acceleartion, as the string s of constant length, m1 and m2 will have the same acceleration

The equations of motion are

(m1+m3)a = (m1+m3)g - T

T - m2g = m2a

eliminating T from both

(m1+m3)a = (m1+m3)g - m2(a+g)

              a =      (m1+m3-m2)g/(m1+m3+m2)

                  = (m1-m2)g/m1

                  = m3g/m1

m3 = m1a/g

      = m2a/(g-a) , putting m1= m2+m3 and rearrange