Two bodies of masses 7 kg and 5 kg are connected by a light string passing over a smooth pulley at the edge of the table as shown in the figure. The coefficient of static friction between the surfaces (body and table) is 0.9. Will the mass `m_1` = 7 kg on the surface move? If not what value of `m_2` should be used so that mass 7 kg begins to slide on the table?

Explanation

As shown in the figure, there are four forces acting on the mass `m_1`

(a) Downward gravitational force along the negative y-axis `(m_1 g) `
(b) Upward normal force along the positive y-axis (N)
(c) Tension force due to mass `m_2` along the positive x axis
(d) Frictional force along the negative x-axis Since the mass m, has no vertical motion, `m_1g=N`
Free body diagram for mass `m_1`
To determine whether the mass m, moves on the surface, calculate the maximum static friction exerted by the table on the mass m. If the tension on the mass m, is equal to or greater than this maximum static friction, the object will move.
`f_(s)^(max) = mu_s m_1 g`
`f_(s)^(max) = 0.9 xx 7 xx 9.8 = 61.74 N`
The tension T ` = m_2g = 5 xx 9.8 xx 49N`
` T lt f_(s)^(max)`
The tension acting on the mass `m_1 = T gt f_(s)^(max) , " when " T = mg `
`m_2 = (mu_s m_1 g)/(g) = mu_s m_1`
`m_2 = 0.9 xx 7 = 6.3 kg `
If the mass `m_2` is greater than 6.3 kg then the mass `m_1` will begin to slide. Note that if there is no friction on the surface, the mass `m_1` will move for `m_2` even for just 1 kg. The values of coefficient of static friction for pairs of materials are given. Note that the ice and ice pair have very low coefficient of static friction. This means a block of ice can move easily over another block of ice.