9. A catapult consists of an arm with a holding bucket at each end as in the dia

Question

9. A catapult consists of an arm with a holding bucket at each end as in the diagram below. The arm along with the holding buckets is pivoted at its centre of mass. Initially, being held in position by a cord, the catapult is loaded with a mass M sitting iii the right hand bucket and a mass m sitting in the left hand bucket. The left part of the arm (from the pivot to the mass m) initially makes an angle alpha below the horizontal. When the cord is cut the arm rotates until the left part of the arm reaches a stop at an angle Beta above the horizontal. Mass m in the left, bucket then continues with speed v. Ignore friction at the pivot and air resistance. The moment of inertia of the catapult arm and buckets without added masses is I0 = 1.0kgm^2. If m = 10kg, M = 100kg, a = 3.0m, b = 1.0m, and angles alpha = Beta= 30degree, then the launch speed v of the mass m is closest to [Hint: use conservation of energy]:

Explanation / Answer

In the original configuration at start,

Initial Potential Energy of mass m (PEim) = -mgh

where h = height of the mass m below the horizontal level = a*sin(30 deg)

Similarly, the PE of the mass M, PEiM = M*gH

where H = b*sin(30 deg)

Initial Kinetic Energies of both the masses = 0 <----as they start from rest

where u = initial velocity = 0 m/s2 <---- starts from rest

Now for the final configuration,where the arm stops and the mass m continues with velocity v.

Final P.E of m, PEfm = mga*sin(30 deg)

and PEfM = -Mgb*sin(30deg)

Final KE of m = 0.5*mv^2

By conservation of energy,

Initital Total Energy must be same as final Total energy,

So, PEim +  PEiM + 0 = PEfm +  PEfM + 0.5*mv^2

So,  -mgh + M*gH = mga*sin(30 deg) -Mgb*sin(30deg) + 0.5*mv^2

So, -mga*sin(30) + Mgb*sin(30) = mga*sin(30 deg) -Mgb*sin(30deg) +  0.5*mv^2

So, 2Mgb*sin(30) - 2*mga*sin(30) = 0.5*mv^2

So, 2g*sin(30)*(Mb-ma) = 0.5*mv^2

So, 2*9.8*sin(30 deg)*(100*1 - 10*3) = 0.5*10*v^2

So, v = 11.7 m/s

So, answer is closest to 10 m.s <------answer

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