© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the physical situation
A circular mop of radius $R$ is pressed against the floor with a total normal force $F$, uniformly distributed over its circular surface. The coefficient of friction between the mop and the floor is $\mu$. The mop is rotated about its central axis at a constant angular speed. We want to find the total torque required to overcome friction.
Step 2: Express the normal force distribution
The total normal force $F$ is uniformly distributed over the area of the circular mop. The area of the mop is $\pi R^2$. Hence, the normal force per unit area is
$$
\frac{F}{\pi R^2}.
$$
Step 3: Consider a small ring element
To find the total torque due to friction, we split the mop into infinitesimal rings of radius $r$ and thickness $dr$. The circumference of this ring is $2\pi r$, and its area is
$$
dA = 2\pi r \, dr.
$$
Step 4: Determine the frictional force on the ring
The normal force on this small ring element is
$$
dN = \left(\frac{F}{\pi R^2}\right) \bigl(2\pi r \, dr \bigr)
= \frac{2F \, r \, dr}{R^2}.
$$
Since frictional force $dF_{\text{fric}}$ on this ring is given by $ \mu \times dN $, we have
$$
dF_{\text{fric}} = \mu \, dN = \mu \left(\frac{2F \, r \, dr}{R^2}\right).
$$
Step 5: Write the torque contribution from the ring
The torque $d\tau$ due to the ring at radius $r$ is
$$
d\tau = \bigl(\text{frictional force at }r \bigr) \times r
= \left(\mu \frac{2F \, r \, dr}{R^2}\right) r
= \mu \frac{2F \, r^2 \, dr}{R^2}.
$$
Step 6: Integrate to find the total torque
To find the total torque $\tau$, integrate $d\tau$ from $r = 0$ to $r = R$:
$$
\tau = \int_{0}^{R} \mu \frac{2F \, r^2}{R^2} \, dr
= \mu \frac{2F}{R^2} \int_{0}^{R} r^2 \, dr.
$$
Evaluate the integral:
$$
\int_{0}^{R} r^2 \, dr = \left[\frac{r^3}{3}\right]_{0}^{R} = \frac{R^3}{3}.
$$
Hence,
$$
\tau = \mu \frac{2F}{R^2} \cdot \frac{R^3}{3}
= \mu \frac{2F R}{3}.
$$
Step 7: Final expression for the torque
The torque required is
$$
\tau = \frac{2}{3}\,\mu\,F\,R.
$$
This matches the given correct answer.
