Physics

(a)

Moment of Inertia of the man-platform system = 7.6 kg-m²

Moment of inertia when the man stretches his hands to a distance 90 cm

= 2 x mr² = 2 x 5 x (0.9)2 = 8.1 kg-m²

Initial moment of inertia of the system,  $-$ = 7.6 + 8.1 = 15.7 kg-m²

Angular speed $-$ = 30 rev/min

Angular momentum,  $-$ = $I_i$ = 15.7 x 30 …… (i)

Moment of inertia when the man folds his hands to a distance of 20 cm

= 2 x mmr²= 2 x 5 x (0.2)2 = 0.4 kg-m²

Final moment of inertia,  ${\omega }_i$ = 7.6 + 0.4 = 8 kg-m²

Final angular speed = $L_i$ and final angular of momentum,  $I_i{\omega }_i$ = $I_f$ = 8 ${\omega }_f$ …… (ii)

From the conservation of angular momentum, we have $L_f$ = $I_f{\omega }_f$

${\omega }_f$ = $I_i{\omega }_i$ = 58.88 rev/min

(b) Kinetic energy is not conserved in the given process. In fact, with the decrease in the moment of inertia, kinetic energy increases. The additional kinetic energy comes from the work done by the man to fold his hands towards himself.

The given situation can be shown as

NB = force exerted on the ladder by the floor point B

NB = force exerted on the ladder by the floor point B

T = Tension in the rope

BA = CA = 1.6 m, DE = 0.5 m and BF = 1.2 m

Mass of the weight, m = 40 kg

The perpendicular drawn from point A on the floor BC, this intersects DE at mid-point H.

Δ ABI and ΔACI are congruent. Therefore BI = IC, I is the mid-point of BC. DE is parallel to BC.

BC = 2 x DE = 1 m and AF = BA – BF = 0.4 m…… (i)

D is the mid-point of AB, hence we can write

AD = (1/2) x BA = 0.8 ……. (ii)

Using equation (i) and (ii), we get FE = 0.4 m

Hence, F is the mid-point of AD.

Hence G will also be the midpoint of AH.

ΔAFG and ΔADH are similar

FG = (1/2)DH = (1/2) X 0.25 = 0.125 M

In ΔADH, AH = $\frac{3AB}{4g\sin ?\theta })$ - $\surd \frac{11.46}{19.6}$ ) = $\frac{FG}{DH}=\frac{AF}{AD}=\frac{FG}{DF}=\frac{0.4}{0.8}=\frac{1}{2}$ – $\surd ({AD}^2$ ) = 0.76 m

For translational equilibrium of the ladder, the upward force should be equal to the downward force, NC + NB = mg = 392 …… (iii)

For rotational equilibrium of the ladder, the net moment about A is

${DH}^2$ NB x BI +mg x FG + NC x CI + T x AG – T x AG = 0

$\surd \left\{\right({0.8}^2$ NB x 0.5 +40 x g + 0.125 x NC x 0.5 = 0

(NC - NB) x 0.5 = 49

NC - NB = 98 ………. (iv)

Adding equation (iii) and (iv) we get

NC = 245 N, NB = 147 N

For rotational equilibrium of the side AB, considering the moment about A

${0.25}^2$ NB x BI + mg x FG + T x AG = 0

$-$ 245 x 0.5 + 40 x 9.8 + T x AG = 0

T = 96.7 N

Initial velocity of the cylinder, v = 5 m/s

Angle of inclination, $\omega$ = 30 $\surd \left(\frac{2}{3}\right)$

Height reached by the cylinder = h

Energy of the cylinder at point A:

KE rot + KEtrans

(1/2) I $\omega$ + (1/2) m $\theta$

Energy of the cylinder at point B = mgh

Using the law of conservation of energy

(1/2) I $°$ + (1/2) m ${\omega }^2$ = mgh

Moment of inertia of the solid cylinder I = (1/2) mr²

Hence (1/2)(1/2)mr² $v^2$ + (1/2) m ${\omega }^2$ = mgh

We also know v = r $v^2$

(1/4)m ${\omega }^2$ + (1/2) m $v^2$ = mgh

(3/4) $\omega$ = gh

h = (3/4)( $v^2$ = (3/4)(25/9.81) = 1.91 m

In Δ ABC, $v^2$ = $v^2$ , AB = BC / $v^2/g)$ = 3.82 m

Hence the cylinder will travel 3.82 m on the inclined plane

For radius of gyration K, the velocity of the cylinder at the instance when it rolls back to bottom is given by the relation

v = $\sin ?\theta$ = $\frac{BC}{AB}$ )

for solid cylinder, $\sin ?30°$ = $\surd \left(\frac{2gh}{1+\frac{k^2}{R^2}}\right)$

v = $\surd (\frac{2gAB\sin ?\theta }{1+\frac{k^2}{R^2}}$ ) = $k^2$ g AB $\frac{R^2}{2}$ )

The time taken to return to the bottom is :

t = $\surd (\frac{2gAB\sin ?\theta }{1+\frac{1}{2}}$ = AB/ $\surd (\frac{4}{3}$ ( $\sin ?\theta$ = $\frac{AB}{v}$ ( $\surd$ = $\frac{4}{3}gAB\sin ?\theta )$ = 0.764 s

Therefore, the total time taken by the cylinder to return to the bottom is 2 x 0.764 = 1.53 s