3. (15 Pts) The uniform crate has a mass of 50 kg and
rests on the cart having an inclined surface. Determine
the smallest acceleration that will cause the crate either
to tip or slip relative to the cart. Will the crate tip or
slip first? The coefficient of static friction between the
crate and the cart is $\mu_s$ = 0.5.
SOLUTION
Equations of Motion: Assume that the crate slips, then $F_f = \mu_s N = 0.5N$.
$\sum M_A = \sum (M k)_A$;
$50(9.81)\cos 15^\circ (x) - 50(9.81)\sin 15^\circ (0.5)$
$= 50a \cos 15^\circ (0.5) + 50a \sin 15^\circ (x)$
(1)
$\sum F_y = m(a_c)_y$;
$N - 50(9.81)\cos 15^\circ = -50a \sin 15^\circ$
(2)
$\sum F_x = m(a_c)_x$;
$50(9.81)\sin 15^\circ - 0.5N = -50a \cos 15^\circ$
(3)
Solving Eqs. (1), (2), and (3) yields
$N = 447.81 \text{ N}$ $x = 0.250 \text{ m}$
$a = 2.01 \text{ m/s}^2$
Ans.
Since $x < 0.3 \text{ m}$, then crate will not tip. Thus, the crate slips.
Ans.
