A cart weighing 200 N can roll without friction along a horizontal path . The cart carries a block weighing 20 N . The coefficient of friction between the block and the cart is `0.25` and g= `10 m//s^(2)` . Find the force of friction between the block and cart and their acceleration when a force of (a) 2N is applied to the block , (b) 20 N is applied to the block .

Explanation

In this problem the mass of block is (20/10) = 2 kg , while of cart (200/10) = 20 kg . The force of limiting friction between block and cart ,
`f_(L) = mu R = mu "mg" `
`= 0.25 xx 20 = 5 N`
(a) When F = 2 N , `" " F lt f_(L)`
So, both will move together with acceleration
`a_(B) = a_(C) = (2)/(20 + 2) = (1)/(11) = 0.09 m // s^(2)` and force of friction will be equal to applied force (and not `muR`) but `Ma = (20)/(11) = 1.8 N ( lt F)`.
(b) When F = 20 N, the block and cart will move with different acceleration as `F gt f_(L)` . For the motion of block ,
`F - f_(L) = ma_(B)`
i .e., `" " 20 -5 = 2a_(B) , i.e., a_(B) = (15//2) = 7.5 m//s^(2)` and for the motion of cart , `f_(L) = Ma_(C)` , i.e., `a_(C) = (5//20) = 0.25 m//s^(2)` and in this situation force of friction will be `f_(L) = 5 N` .