Fall 2007 2 Mech

A wedge of mass M rests on a horizontal frictionless surface. A point mass m is placed on the wedge, whose surface is also frictionless. Find the horizontal acceleration a of the wedge.

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Let N be the normal reaction force, then the Newton equation for the wedge projected on the horizontal axis gives

(1)
\begin{align} Ma = N \sin{\alpha} \end{align}
(2)
\begin{align} N = Ma/ \sin{\alpha} \end{align}

Consider now the motion of the small mass. Its acceleration is $\bold{a_\perp} + \bold{a_{||}}$. The second term $\bold{a_{||}}$ is along the incline, and so it vanishes if projected on the direction normal to the incline. The corresponding Newton law reads:

(3)
\begin{align} -m a \sin{\alpha} = N - mg \cos{\alpha} \end{align}

Substituting N from the first equation and solving for a, we get

(4)
\begin{align} a = g \frac{\sin{\alpha} \cos{\alpha}}{M/m + \sin^2{\alpha}} \end{align}