A disc rolls over a horizontal floor without slipping with a linear speed of 5 cm/sec. Then the linear speed of a particle on its rim, with respect to the floor, when it is in its highest position is?

Options

A. 10cm/sec
B. 5cm/sec
C. 2.5cm/sec
D. 0

Explanation

To find the linear speed of a particle on the rim of a disc when it is at its highest position while rolling without slipping, we can follow these steps:

### Step 1: Understand the motion of the disc
The disc is rolling on a horizontal surface with a linear speed (V) of 5 cm/sec. In rolling motion, the center of mass of the disc moves forward, and all points on the disc have different velocities depending on their position relative to the center of mass.

**Hint:** Remember that in rolling motion, the speed of the center of mass is the same for all points on the disc, but rotational motion adds to the speed of points on the rim.

### Step 2: Identify the point of interest
We are interested in the speed of a particle located at the highest point of the disc. At this position, the particle has both translational and rotational motion.

**Hint:** The highest point of the disc is where the particle is farthest from the center of mass in the upward direction.

### Step 3: Calculate the translational speed
The translational speed (V) of the center of mass is given as 5 cm/sec. This speed applies to all points on the disc, including the highest point.

**Hint:** The translational speed is constant and equal to the speed of the center of mass.

### Step 4: Calculate the rotational speed
For a point on the rim of the disc, the rotational speed (Vr) can be calculated using the relationship:
\[ V_r = R \cdot \omega \]
where \( R \) is the radius of the disc and \( \omega \) is the angular velocity.

Since the disc rolls without slipping, we have:
\[ V = R \cdot \omega \]
Thus:
\[ \omega = \frac{V}{R} \]

**Hint:** The angular velocity can be derived from the linear speed and radius of the disc.

### Step 5: Find the total speed at the highest point
At the highest point, the total speed (Vp) of the particle is the sum of the translational speed and the rotational speed:
\[ V_p = V + V_r \]
Since \( V_r = V \) (because at the highest point, the rotational speed adds to the translational speed):
\[ V_p = V + V = 2V \]

### Step 6: Substitute the values
Substituting the value of V:
\[ V_p = 2 \times 5 \, \text{cm/sec} = 10 \, \text{cm/sec} \]

### Conclusion
The linear speed of the particle on the rim of the disc at its highest position is **10 cm/sec**.

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