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METACENTRIC HEIGHT AND RADIUS RADIUS OF GYRATION OF FLOATING BODIES
OBJECTIVE To determine metacentric height and radius of gyration of the given floating body. THEORY Metacentre: It is defined as the point about which a body starts oscillating when it is tilted by a small angle. The metacentre may also defined as the point at which line of action of the point of buoyancy will meet the normal axis of the body when the body is given a small angular displacement. Metacentric height: the distance between the metacentre of a floating body and the centre of gravity of the body is called metacentric height. Radius of gyration: It is defined as the distance of the centre of gravity of the body from a reference axis. For a body to be equilibrium W= F b and both weight of the body(W) and the buoyant force(Fb) are acting along the same vertical line. When a body is tilted through a small angle Θ (known as angle of heel) by two movable weights placed across the deck the centre of buoyancy shifts from B to B 1 to the right and there is a parallel shift of the total centre of gravity of the body, i.e. centre of gravity of the body including movable weights. If a vertical line is drawn through the new position of the buoyant force ie through point B 1, it will intersect the initial line of action of buoyant force through point B at point M. The point M is the metacentre and the distance GM,metacentric height. The metacentric height gives a measure of stability for a floating body.
The metacentric height is obtained by equating the moment due to movement of movable weights and the moment due to shifting of G to G 1 and is given by the following expression.s w1x-w2x2
w1x1-w2x2 GM =
Where
Wtan Θ
w1 and w2 are the movable weights and x1 & x2 are their respective distances from the centre of the cross bar. W is the total weight of the floating body including the movable weights, Θ is the angle of tilt.
Time period of oscillation of a floating body is given by T= 2 π (k2/gh)1/2 K=radius of gyration of the body about longitudinal axis passing through the centre of gravity. h = metacentric height, g=acceleration due to gravity APPARATUS A float tank and ship model with pendulum and protractor for tilt measurement. A set of movable weights and stopwatch for time period measurement. PROCEDURE 1 Measure the dead weight of the model. 2 Adjust the pointer to zero reading when no tilting moment is applied by adjusting the balancing weights provided at the ends of cross bar. 3 Displace the movable masses across the bar so as tilt the model through a angle.
4 Note the distances of the known movable weights from the centre of the cross bar and the tilt angle Θ. 5 Repeat the process for five different angle of tilt by by varying the movable weight position. 6 Bring the float to balanced position without the loads. 7 Allow the float to oscillate and note the time taken for five continuous oscillations. 8 Repeat the procedure and calculate the average time period.
APPLICATION Floating vessels design
RESULT
INFERENCE
SAMPLE CALCULATION 1 Dead mass of float(W) Base area of float tank(A)=
W d= Vρ
Initial level of water in tank(h 1)=
=
Final level of water(with float)( h 2)=
=
kg
volume of water displaced(V)=A(h displaced(V)=A(h 2-h1) = Density of water( ρ)
=
2 Disturbing moment(wx)
wx= w1x1-w2x2
Disturbing mass( w1)= Disturbing mass( w2)= Distance of w1 from centre(x1)= Distance of w2 from centre(x2)=
3 Metacentric height(h) Disturbing moment (wx)=
wx
H =
Total mass of float(W)= Wd+ masses added
Wtan Θ =
= =
cm
Angle of tilt( Θ)
=
4 Radius of gyration(K) Time period of oscillation(T)= oscillation(T)= 1/2
K= T(gh) /2π
Metacentric height(h)= Acceleration due to gravity(g)=