A cart of mass M is on a smooth horizontal table. The top surface of the cart is combined of an incline 
and a horizontal plane 
. The center of mass of the cart is at C. On this cart, there is a mass m at A (see figure.) The two masses are released from rest. The mass m slides with no friction on the top surface of the cart. Assume that the corner B is rounded so that m passes it smoothly. Further assume that H, L, λ , θ , d, h and g are known in this problem.
How long does it take for m to slide from B to D?
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. We will first find the velocities of m and M at the moment m arrives at B. Here we can use the law of conservation of momentum since there are no external forces, and also conservation of energy since there is no friction. By the law of conservation of momentum, we have:
mv + Mv ′ = 0 …...(i)
while the law conservation of energy implies
mv 2 + 
mv ′ 2 = mgH …...(ii)
v is the velocity of m when it arrives at B and v ′ is the velocity of M at B. From Eq. (i) we have v = 
, and by plugging it into Eq. (ii) we obtain
v ′ = 
…...(iii)
Therefore, v = – 
= – 
…...(iv)
v is the velocity of m relative to the ground. The velocity relative to the cart is 
, where 
= v– v ′ . Therefore, the time required is :
t = 
= 
= 
…...(v)
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