Answer :
It takes the cylinder approximately 0.82 seconds to travel 1.59 m down the incline.
Here's how to find the time it takes the cylinder to travel 1.59 m down the incline:
**1. Analyze the forces:**
- **Weight (mg):** Acts downward with a magnitude of mg = 1.61 kg * 9.81 m/s² ≈ 15.85 N.
- **Normal force (N):** Acts perpendicular to the incline and balances the component of the weight normal to the incline.
- **Friction force (f):** Zero due to the condition of rolling without slipping.
- **Gravitational force parallel to the incline (mg sinθ):** Acts down the incline with a magnitude of mg sinθ = 15.85 N * sin(21.9°) ≈ 5.48 N.
**2. Apply the rotational and translational equations:**
- **Rotational equation:** Torque due to gravity = moment of inertia * angular acceleration (α).
τ_g = I_cm * α = (1/2) * M * r² * α
- **Translational equation:** Net force down the incline = mass * acceleration (a).
F_net = ma = mg sinθ - f = ma
**3. Combine the equations:**
Since f = 0, we can rewrite the translational equation:
ma = mg sinθ
Substitute the rotational equation for α:
ma = mg sinθ
ma = (1/2) * M * r² * (a/r)
2a = (M/I_cm) * a
Solve for a:
a = (2 * g * sinθ) / (1 + (M * r²) / I_cm)
a ≈ 3.16 m/s²
**4. Calculate the time:**
Use the kinematic equation for distance and zero initial velocity:
d = 1/2 * a * t²
t = sqrt(2d / a)
t = sqrt(2 * 1.59 m / 3.16 m/s²) ≈ 0.82 s
Therefore, it takes the cylinder approximately 0.82 seconds to travel 1.59 m down the incline.
The probable question may be:
A cylinder (r = 0.143 m, Icm = 2.420×10-2 kg m2, M = 1.61 kg) starts from rest and rolls without slipping down a plane with an angle of inclination of θ = 21.9 deg. Find the time it takes it to travel 1.59 m along the incline.
