Q4. A disc of radius 0.2 m rolls without slipping, and its centre moves at 3 m s. Find the speed of the point halfway between the centre and the top of the disc.
The angular speed is ω = v/R = 3/0.2 = 15 rad s. A point 0.1 m above the centre moves forward at v from the translation and forward at ωr = 15 × 0.1 = 1.5 m s from the rotation, so its speed is 3 + 1.5 = 4.5 m s. Check: the point is 0.3 m above the contact point, and 15 × 0.3 = 4.5 m s.
Q5. A car moving at 10 m s brakes so hard that its wheels lock and stop turning. Just after the wheels lock, what kind of motion does a wheel have, and how fast does its contact point slide over the road?
With ω = 0 the wheel is in pure translation: every point, including the contact point, moves forward at 10 m s. At that moment the contact point slides over the road at 10 m s (less as the car slows), so the friction on the tyre is kinetic.
Q6. A heavy book is (a) pushed across a table so that it does not turn, and (b) tossed so that it spins as it flies. In which case does a line drawn along one edge of the book keep its direction? What kind of motion is each?
In (a) the line keeps its direction: the book is in pure translation. In (b) the book translates and rotates together. An edge generally changes direction, but an edge parallel to a fixed rotation axis can keep its direction even while the book spins. (NCERT suggests trying both motions with a heavy book.)
Q7. A wheel rolls without slipping with its centre moving at v. Show that the speed of a rim point that has turned through 90° from the contact point is √2 times the speed of the centre.
At that point the translation gives v forward, and the rotation gives ωR along the rim, which there is vertical. Without slipping ωR = v, so the two parts are equal and at right angles: the speed is √(v + v) = √2 v.