At the bottom, the total energy is converted into translational and rotational energy. The translational energy is 2/3 of the total energy.

At the bottom, the total kinetic energy is split equally between translational and rotational for a disk, hence the ratio is 1:1.

Using conservation of energy, potential energy at height h converts to kinetic energy at the bottom. The speed is √(2gh).

The relationship between linear speed and angular speed for rolling without slipping is given by ω = v/R.

Angular momentum L = Iω, and if the radius is doubled, the moment of inertia I becomes 4I, thus L = 4Iω.

To conserve angular momentum, if the radius is doubled, the angular velocity must be halved.

The linear velocity v = rω. If the radius is doubled, to maintain the same v, the angular velocity must remain ω.

The moment of inertia I of a disk is proportional to r^2, so if the radius is doubled, I becomes 4I. Thus, angular momentum L = Iω becomes 4Iω.

The moment of inertia of a disk is I = (1/2)MR^2. If the radius is doubled, the new moment of inertia becomes I' = (1/2)M(2R)^2 = 4I.

Angular momentum L = Iω. If the radius is doubled, the moment of inertia increases by a factor of 4, thus L = 4Iω.

The absolute error is given as 2 m.

Using the lens formula 1/f = 1/v - 1/u, we have 1/(-10) = 1/v - 1/5. Solving gives v = -3.33 cm.

A diverging lens always forms a virtual and upright image when the object is placed in front of it.

Torque (τ) = Force (F) × Distance (r) = 20 N × 0.8 m = 16 Nm.

Torque (τ) = F × d = 20 N × 1 m = 20 Nm.

Torque = Force × Distance = 50 N × 1.2 m = 60 Nm.

Torque (τ) = Force (F) × Distance (r) = 50 N × 1 m = 50 Nm.

Using the lens maker's formula, R = 2f(n-1) = 2*10*(1.5-1) = 20 cm.

Using the lens maker's formula, f = R/(n-1), we find f = 20/(1.5-1) = 40 cm.

For a drop to remain in equilibrium, the upward force due to surface tension must balance the downward gravitational force.

The spherical shape of a liquid drop is due to surface tension, which minimizes the surface area for a given volume.

The drop of oil spreads on water because the surface tension of water is higher than that of oil, causing the oil to spread out.

A drop of water is spherical due to surface tension, which minimizes the surface area.

Daily production = 2000 items / 5 days = 400 items/day. In 12 days, it produces 400 × 12 = 4800 items.

The possible combinations of children are BB, BG, GB, GG. Given that at least one is a boy, we can eliminate GG, leaving us with BB, BG, GB. Out of these 3 combinations, only 1 is BB. Therefore, the probability is 1/3.

The possible combinations are BB, BG, GB. Given at least one is a boy, the combinations are BB, BG, GB. The probability of both being boys is 1/3.

The possible combinations of children are BB, BG, GB, GG. Out of these, only 1 combination is both boys (BB). Thus, the probability is 1/4.

The only combinations with at least one boy are: BBB, BBG, BGB, GBB, BGG, GBG, GGB. Out of these, all combinations except BBB have at least one girl. Thus, P(At least one girl | At least one boy) = 6/7.

The only scenario where there are no girls is if all are boys. The probability of at least one girl given at least one boy is 3/4.

Total income = Average income × Number of members = 5000 × 4 = $20000.