Determine the height h so that the particle with mass m just

Determine the height h so that the particle with mass m just has contact at the top of the loop with radius R in the cases: point particle (slides without friction) Sphere of radius r where R = 5r (rolling without sliding)

Solution

>> Case (a), [ Point Particle ]

>> Now, for it to not loose contact at top point of loop of radius R,

Let\'s assume Velocity of particle at top point of loop = \"V\"

Now, At to point of loop, on particle ,

forces acting are :

1). Weigt, W = mg

2). Normal reaction between contact, N

3). Centripetal Force, Fc = mV²/R

>> As, Fc = N + W

and for just not loose the contact, => N = 0

=> Fc = W

=> mV²/R = W ...............(1)..........

>> Now, Applying energy conservation at top point of loop and at height h above the plane

=> (1/2)mV² + mg(2R) = mgh

From (1)....

WR/2 + 2mgR = mgh

As, W = mg

=> 2.5*mgR = mgh

=> h = 2.5 R .....ANSWER..........

>> Case (2) .....Sphere of radius r

>> Here, also, upto equation (1), will hold true

>> Now, Energy at toppoint of loop = Kinetic Energy ( due to trnaslational + due to rotational )

=> Energy at top point of loop = (1/2)mV² + (1/2)Iw²

Now, I = Moment of Inertia of sphee = (2/5)mr²

and, as w = V/r

=> Energy = (1/2)mV² + (1/2)(2/5)mr²(V/r)² = (7/10) mV²

>> Now, applying energy conservation at top point of loop and at height h above the plane ,

=> (7/10)mV² + mg(2R + r) = mgh

From (1) and using W = mg and r = R/5 = 0.2*R

=> 0.7*mgR + 2.2*mgR = mgh

=> h = 2.9*R ..........REQUIRED ANSWER............