Ncert Exercises & Solutions

Q.18 A uniform sphere of mass m and radius R is placed on a rough horizontal surface (figure). The sphere is struck horizontally at a height h from the floor. Match the following

(a) h = R/2 (i) Sphere rolls without slipping with a constant velocity and no loss of energy.

(b) h = R (ii) Sphere spins clockwise, loses energy by friction.

(d) h = 7R/5 (iv) Sphere has only a translational motion, loses energy by friction.

Hint: Apply conservation of angular momentum.

Step 1: Find h by applying angular momentum conservation.

Consider the diagram where a sphere of m and radius R, struck horizontally at height h above the floor

The sphere will roll without slipping when

ω = \frac{v}{r}

where, v is linear velocity and

ω

is the angular velocity of the sphere.

Now, the angular momentum of the sphere, about center of mass

[We are applying conservation of angular momentum just before and after struck]

\begin{array}{l} mv (h - R) = Iω = (\frac{2}{5} {mR}^{2}) (\frac{v}{R}) \\ \Rightarrow mv (h - R) = \frac{2}{5} mvR \\ h - R = \frac{2}{5} R \Rightarrow h = \frac{7}{5} R \end{array}

Therefore, the sphere will roll without slipping with a constant velocity and hence, no loss of energy, so

The torque due to the applied force, F about the center of mass

τ

= F(h - R) (clockwise)

For Q h R, the sphere will have only translational motion. It would lose energy through friction.

Hence, (b)

\to

(iv)

The sphere will spin clockwise when

τ > 0 \Rightarrow h > R

Therefore.
(C)

\to

(ii)
The sphere will spin anti-clockwise when

τ < 0 \Rightarrow h < R, (a) \to (ii)

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