Laws of Motion

One end of a massless spring of spring constant k and natural length ${\mathcal{l}}_0$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ <!-- [if gte mso 9]><xml> </xml><![endif]--> about an axis passing through fixed end, then the elongation of the spring will be

5.40 A thin circular loop of radius R rotates about its vertical diameter with an angular frequency $\mathit{\omega }$ . Show that a small bead on the wire loop remains at its lowermost point for $\le \sqrt{\mathit{g}/\mathit{R}}$ . What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for $\mathit{}\mathit{\omega }$ = $\sqrt{2\mathit{g}/\mathit{R}}$ ? Neglect friction.

5.39 A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 200 rev/min. The coefficient of friction between the wall and his clothing is 0.15. What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed ?

5.38 You may have seen in a circus a motorcyclist driving in vertical loops inside a death well’ (a hollow spherical chamber with holes, so the spectators can watch from outside).

Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m ?

5.37 A disc revolves with a speed of $33\frac{1}{3}$ rev/min, and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the co-efficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record ?

5.36 The rear side of a truck is open and a box of 40 kg mass is placed 5 m away from the open end as shown in Fig. 5.22. The coefficient of friction between the box and the surface below it is 0.15. On a straight road, the truck starts from rest and accelerates with 2 m s^-2. At what distance from the starting point does the box fall off the truck? (Ignore the size of the box).

5.35 A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley is 0.18. The trolley accelerates from rest with 0.5 m s^-2 for 20 s and then moves with uniform velocity.

Discuss the motion of the block as viewed by

(a) A stationary observer on the ground

(b) An observer moving with the trolley

5.34 Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall (Fig. 5.21). The coefficient of friction between the bodies and the table is

0.15. A force of 200 N is applied horizontally to A. What are

(a) The reaction of the partition

(b) The action-reaction forces between A and B ? What happens when the wall is removed?

Does the answer to (b) change, when the bodies are in motion?

Ignore the difference between μs and μk

5.33 A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey

(a) Climbs up with an acceleration of 6 m s^-2

(b) Climbs down with an acceleration of 4 m s^-2

(c) Climbs up with a uniform speed of 5 m s^-1

(d) Falls down the rope nearly freely under gravity?

(Ignore the mass of the rope)

5.32 A block of mass 25 kg is raised by a 50 kg man in two different ways as shown in Fig. 5.19. What is the action on the floor by the man in the two cases ? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding ?