ANKUSH SACHDEVA IS RIGHT AND PROVE OF CENTRE OF mass of cone is posted below by me!!!!!! The radius r of a horizontal slice of the cone of thickness dz through (0, 0, z) is given by: r^2 = x^2 + y^2
Its volume dV is: dV = pi r^2 dz ...(1)
If H is the height of the cone, and R its base radius, then from similar triangles: H / R = (H - z) / r r = R(H - z) / H
Substituting for r in (1): dV = pi R^2 (H - z)^2 / H^2 dz
The moment dM of this slice about the base is: dM = z dV = (pi R^2 / H^2) z(H - z)^2 dz
If C is the height above the base of the centre of mass, and V is the volume of the cone, then the moment of the whole volume is: M = CV = C pi R^2 H / 3 ...(2)
Equating the values of M from (1) and (2): C pi R^2 H / 3 = pi R^2 H^2 / 12 C = H / 4.
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ReplyDeletesqrt [2g/L]
ReplyDeletewe had to find the position of c.o.m. of cone (though i just googled it) to solve this,otherwise its easy
ReplyDeletefR/2 = mgR (R/2 is the ht of com form base)
ReplyDeletef=mw^2L
put it in eqn 1
get w = root 2g/L
ANKUSH SACHDEVA IS RIGHT AND PROVE OF CENTRE OF mass of cone is posted below by me!!!!!! The radius r of a horizontal slice of the cone of thickness dz through (0, 0, z) is given by:
ReplyDeleter^2 = x^2 + y^2
Its volume dV is:
dV = pi r^2 dz ...(1)
If H is the height of the cone, and R its base radius, then from similar triangles:
H / R = (H - z) / r
r = R(H - z) / H
Substituting for r in (1):
dV = pi R^2 (H - z)^2 / H^2 dz
The moment dM of this slice about the base is:
dM = z dV
= (pi R^2 / H^2) z(H - z)^2 dz
The moment M of the whole cone is:
M = pi int (0, H) [ z(H - z)^2 ] dz
= (pi R^2 / H^2) int (0, H)[ z(H - z)^2 ] dz
= (pi R^2 / H^2) int (0, H) [ zH^2 - 2Hz^2 + z^3 ] (0, H)
= (pi R^2 / H^2) [ H^2 z^2 / 2 - 2Hz^3 / 3 + z^4 / 4 ](0, H)
= (pi R^2 / H^2) [ H^4 / 2 - 2H^4 / 3 + H^4 / 4 ]
= (pi R^2 / H^2) [ H^4 (1 / 2 - 2 / 3 + 1 / 4) ]
= (pi R^2 H^2) / 12 ...(1)
If C is the height above the base of the centre of mass, and V is the volume of the cone, then the moment of the whole volume is:
M = CV
= C pi R^2 H / 3 ...(2)
Equating the values of M from (1) and (2):
C pi R^2 H / 3 = pi R^2 H^2 / 12
C = H / 4.
root(2g/l)
ReplyDeleteyess. sqrt(2g/L) is the right answer...
ReplyDeletetry some pseudo force people.. you'll get a good feeling of what's happenin..
ReplyDeleteanuj,bohot fight mar li yaar..COM ka distance frm base k bina nai ho rha..is there a way to do it without that?
ReplyDeletena..ho hi nahi sakta.. the maxm. angular velocity is a property of the object kept and the distance from the axis..
ReplyDeleteaur yaar pseudo force bhi to COM se hi paas karegi!!!torque nikalne ke liye you need the distance!
ReplyDelete