A cart of mass M rests on a frictionless horizontal surface, and a...

Since there is no external force acting on the system (cart and bob) in the horizontal direction, the horizontal position of the center of mass of the system remains stationary.

Let $x$ be the displacement of the cart relative to the ground. The displacement of the bob relative to the cart is $L$.

Therefore, the displacement of the bob relative to the ground is $x + L$.

Using the principle of conservation of the center of mass position:

$M \Delta x_{M} + m \Delta x_{m} = 0$

$M x + m (x + L) = 0$

$(M + m) x + mL = 0$

$x = -\dfrac{mL}{M + m}$

The negative sign indicates that the cart moves in the direction opposite to the displacement of the bob relative to the cart. The magnitude of the displacement of the cart is $\dfrac{mL}{m + M}$.