A hollow cylinder is rolling on an inclined plane, inclined at an angle of `30^(@)` to the horizontal. Its speed after travelling a distance of 10 m will be

Options

A. 49 m/sec
B. 0.7 m/sec
C. 7 m/sec
D. Zero

Answer

To solve the problem of finding the speed of a hollow cylinder rolling down an inclined plane after traveling a distance of 10 m, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Angle of inclination, θ = 30°
- Distance traveled along the incline, d = 10 m
- Acceleration due to gravity, g = 9.8 m/s²

2. **Determine the Height Change:**
- The height (h) can be calculated using the sine of the angle of inclination:
\[ h = d \cdot \sin(\theta) = 10 \cdot \sin(30°) = 10 \cdot \frac{1}{2} = 5 \text{ m} \]

3. **Apply Conservation of Energy:**
- The potential energy (PE) lost by the cylinder as it rolls down is converted into kinetic energy (KE).
- The potential energy at the height h is given by:
\[ PE = mgh \]
- The total kinetic energy when the cylinder is rolling is the sum of translational kinetic energy (TKE) and rotational kinetic energy (RKE):
\[ KE = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \]
- For a hollow cylinder, the moment of inertia (I) is given by:
\[ I = mr^2 \]
- The relationship between linear velocity (v) and angular velocity (ω) is:
\[ \omega = \frac{v}{r} \]
- Substituting this into the kinetic energy equation gives:
\[ KE = \frac{1}{2} mv^2 + \frac{1}{2} \left(mr^2\right) \left(\frac{v}{r}\right)^2 = \frac{1}{2} mv^2 + \frac{1}{2} mv^2 = mv^2 \]

4. **Set Up the Energy Equation:**
- According to the conservation of energy:
\[ mgh = mv^2 \]
- Canceling the mass (m) from both sides:
\[ gh = v^2 \]

5. **Substituting Values:**
- Substitute g = 9.8 m/s² and h = 5 m into the equation:
\[ 9.8 \cdot 5 = v^2 \]
\[ 49 = v^2 \]

6. **Calculate the Speed:**
- Taking the square root of both sides to find v:
\[ v = \sqrt{49} = 7 \text{ m/s} \]

### Final Answer:
The speed of the hollow cylinder after traveling a distance of 10 m down the inclined plane is **7 m/s**.

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