Problem 1: Find the Mass of the Block Hanging from a Pulley
Given:
- Radius of pulley, r = 42.23 cm = 0.4223 m
- Mass of pulley, mp = 6.779 kg
- Acceleration of block, a = 4.325 m/s²
- Gravitational acceleration, g = 9.8 m/s²
To Find: Mass of the block, mb
Solution:
Torque, τ = Iα, where I = mpr²
Tension, T = mpa
Net force: mbg − T = mba
Substitute T into the equation:
mbg − mpa = mba
Rearranging the terms:
mb (g − a) = mp a
Solve for mb:
mb = (mp × a) / (g − a) = (6.779 × 4.325) / (9.8 − 4.325)
Mass of the block ≈ 5.36 kg
Problem 2: Distance to Place Cup to Catch a Rolling Ball on an Inclined Board
Given:
- Length of board, L = 1.43 m
- Width/Height of support, H = 12.3 cm = 0.123 m
To Find: Horizontal distance d from elevated end where the cup should be placed
Solution:
Determine the angle of inclination using tangent function:
tan θ = H / L = 0.123 / 1.43 ≈ 0.086 → θ ≈ 4.92°
Now, calculate horizontal displacement of the cup:
d = L × sin(θ) = 1.43 × sin(4.92°) ≈ 1.43 × 0.0858
Distance d ≈ 0.123 m
Problem 3: Time to Stop a Rotating Disk System
Given:
- Disk A: m = 2.89 kg, r = 12.5 cm = 0.125 m
- Disk B: m = 0.289 kg, r = 1.25 cm = 0.0125 m
- Torque due to friction, τ = 0.233 Nm
- Initial angular velocity, ω₀ = -6π rad/s
Solution:
Calculate the moment of inertia of both disks:
I = ½ m1 r1² + ½ m2 r2²
= ½ × 2.89 × (0.125)² + ½ × 0.289 × (0.0125)²
≈ 0.0226 + 0.00002 = 0.02262 kg·m²
= ½ × 2.89 × (0.125)² + ½ × 0.289 × (0.0125)²
≈ 0.0226 + 0.00002 = 0.02262 kg·m²
Angular deceleration:
α = τ / I = 0.233 / 0.02262 ≈ 10.3 rad/s²
Using angular motion formula:
0 = ω₀ + αt ⇒ t = -ω₀ / α = 6π / 10.3 ≈ 1.83 s
Time to stop ≈ 1.83 s