Question

A physics student of mass 43.0 kg is standing at the edge of the flat roof of a building, 12.0 m above the sidewalk. An unfriendly dog is running across the roof toward her. Next to her is a large wheel mounted on a horizontal axle at its center. The wheel, used to lift objects from the ground to the roof, has a light crank attached to it and a light rope wrapped around it; the free end of the rope hangs over the edge of the roof. The student radius 0.300 m and a moment of inertia of 9.60 kg m^2 for rotation about the axle, how long does it take her to reach the side walk, and how fast will she be moving just beofre she lands?

Answer 1

The speed of the student just before she lands, v₂ is approximately 8.225 m/s

Step-by-step explanation:

The given parameters are;

The mass of the physic student, m = 43.0 kg

The height at which the student is standing, h = 12.0 m

The radius of the wheel, r = 0.300 m

The moment of inertia of the wheel, I = 9.60 kg·m²

The initial potential energy of the female student, P.E.₁ = m·g·h₁

Where;

m = 43.0 kg

g = The acceleration due to gravity ≈ 9.81 m/s²

h = 12.0 h

∴ P.E.₁ = 43 kg × 9.81 m/s² × 12.0 m = 5061.96 J

The kinetic rotational energy of the wheel and kinetic energy of the student supporting herself from the rope she grabs and steps off the roof, K₁, is given as follows;

`K_1 = (1)/(2) \cdot m \cdot v_(1)^2+(1)/(2) \cdot I \cdot \omega_(1)^2`

The initial kinetic energy, 1/2·m·v₁² and the initial kinetic rotational energy, 1/2·m·ω₁² are 0

∴ K₁ = 0 + 0 = 0

The final potential energy of the student when lands. P.E.₂ = m·g·h₂ = 0

Where;

h₂ = 0 m

The final kinetic energy, K₂, of the wheel and student is give as follows;

`K_2 = (1)/(2) \cdot m \cdot v_(2)^2+(1)/(2) \cdot I \cdot \omega_(2)^2`

Where;

v₂ = The speed of the student just before she lands

ω₂ = The angular velocity of the wheel just before she lands

By the conservation of energy, we have;

P.E.₁ + K₁ = P.E.₂ + K₂

∴ m·g·h₁ +
`(1)/(2) \cdot m \cdot v_(1)^2+(1)/(2) \cdot I \cdot \omega_(1)^2` = m·g·h₂ +
`(1)/(2) \cdot m \cdot v_(2)^2+(1)/(2) \cdot I \cdot \omega_(2)^2`

Where;

ω₂ = v₂/r

∴ 5061.96 J + 0 = 0 +
`(1)/(2) * 43.0 \, kg * v_(2)^2+(1)/(2) * 9.60 \, kg\cdot m^2 \cdot \left ((v_2)/(0.300 \, m) }\right ) ^2`

5,061.96 J = 21.5 kg × v₂² + 53.
`\overline 3` kg × v₂² = 21.5 kg × v₂² + 160/3 kg × v₂²

v₂² = 5,061.96 J/(21.5 kg + 160/3 kg) ≈ 67.643118 m²/s²

v₂ ≈ √(67.643118 m²/s²) ≈ 8.22454363 m/s

The speed of the student just before she lands, v₂ ≈ 8.225 m/s.