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Q5.40 A thin circular loop of radius R rotates about its vertical diameter with an angular frequency \omega. Show that a small bead on the wire loop remains at its lowermost point for \omega \leq \sqrt{g/R}. What is the angle made by the radius vector joining the centre to the bead with the vertically downward direction for \omega = \sqrt{2g/R}? Neglect friction.

The FBD of the loop is given below :                                                  Using equilibrium conditions we can write :                                                 and                                        Consider  OPQ, we have :                                                  or                                               Using l in above equation we get :                  ...

Q5.39 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?

Frequency of rotation is :                                                 The required condition so that man will not fall :                                                or                                          or                                          or                                            And thus :                                                                    Thus the...

Q5.38 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?

The motorcyclist does not drop down when he is at the uppermost point because of their weight and the normal force is balanced by the centripetal force.                                                  or                                              At minimum velocity, the normal reaction is zero. So the equation becomes :                                                              or       ...

Q5.37 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 coefficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record?

Frequency of revolution is :                                                    (i) Case 1 :- When coin is placed at 4 cm :                                            Radius  =  0.04 m                                     Angular frequency :                                                                       or                                                                        The...

Q5.36    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).

Using Newton's second law of motion we can write :                                                                                                                                                   Also, the frictional force is given by :                                                                      or                                                                      Thus net force...

Q5.35 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 (b) an observer moving with the trolley.

(b)  With reference to the observer moving with trolley, the trolley will be at rest as a pseudo force will act to balance out the frictional force. In relative motion both are moving together, thus are rest with respect to each other. 

Q5.35 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.

(a) Using Newton's second law of motion we can write :                                 or                                    Its direction is in the direction of motion of trolley.                           The frictional force is      or                                                              Since the frictional force is greater than the applied force so the block will appear to be at...

Q5.34  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 (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 \mu_s and \mu_k.

    

Consider block A.       The frictional force on block A will be :                                   or                                      Thus net force on B due to block A is  =  200  -  7.5   = 192.5 N. The net force on the partition is 177.5 N. Using Newton's law of motion we have,                                    or                                For block A :                        ...

Q5.34 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? What happens when the wall is removed? Ignore the difference between \mu_s and \mu_k.

            

The applied force is 200 N. The maximum friction force on the system is given by :                                             or                                              So the net force on partition is  200  -  22.5  =  177.5  N. This will be equal to the reaction of the partition (action-reaction pair). The direction will be leftward. 

Q5.33  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).

    

Given that   Tmax  =  600 N. (a)   Acceleration of  6 m s-2 in the upward direction:-                               Using Newton's law of motion we can write  :                                                               or                                                                Thus rope will break. (b)   Acceleration of 4 m s-2 in a downward direction:-                             ...

Q5.32 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?

            

Using Newton's law force applied on the block:-             or                                                                                                                                                              Weight of man   =   500 N Case 1:-  When a man is lifting block directly, man is applying force in the upward direction directly.                            The net force on...

Q5.31 A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is 106 kg. What provides the centripetal force required for this purpose — The engine or the rails? What is the angle of banking required to prevent wearing out of the rail?

The required centripetal force is provided by the rails, as by Newton's third law of motion, wheels apply force on the rails and thus rails provides a force on the wheels (action-reaction pair). We know that the angle of banking is given by :                                                       or                                                                 or                               ...

Q5.30 An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop?

Convert speed of aircraft in SI units :                                                              We are familiar with the following relation :                                                               or                                                                   or                                                                         or                                         ...

Q5.29 Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of

(c) the reaction of the 6th coin on the 7th coin.

Since the 6th coin will experience a force due to 4 coins that are present above i.e., 4 mg. According to Newton's law of action-reaction pair, the 6th coin will have a reaction force on 7th coin of magnitude 4 mg in the upward direction.

Q5.29 Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of

(b) the force on the 7th coin by the eighth coin,

The eighth coin is placed directly above the 7th coin. Thus the net force experienced by 7th coin is due to 8th coin (as the 7th coin is in contact with 8th coin only from the top). Hence the force on the 7th coin by the eighth coin is 3 mg.

Q5.29 Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of
(a) the force on the 7 th coin (counted from the bottom) due to all the coins on its top.

The weight on the 7th coin is due to the top 3 coins. So required force is equal to the weight of 3 coins  =   3 mg This force is acting vertically downward.

Q5.28 A stream of water flowing horizontally with a speed of 15 m s-1 gushes out of a tube of cross-sectional area 10-2 m2, and hits a vertical wall nearby. What is the force exerted on the wall by the impact of water, assuming it does not rebound?

Firstly we will calculate the mass of water passing through per second  :                                                      or                                                          or                                                          Force is defined as the rate of change of momentum.                                                      or                                           ...

Q5.27 A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s-2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the

(c) force on the helicopter due to the surrounding air.

       or                                                  or                                                         or                                                         The direction of the force of the rotor on the surrounding will be in the downward direction. By action-reaction pair, the force on helicopter due to the surrounding is 32500 N and it is directed vertically upward. 

Q5.27 A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s-2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the

(b) action of the rotor of the helicopter on the surrounding air,

(b) In this case, we need to consider helicopter and passengers in a system because we need to determine the action of the rotor.         So by Newton's laws motion, we have :                                                      or                                                  or                                                         or                                                     ...

Q5.27 A helicopter of mass 1000 kg rises with a vertical acceleration of 15ms^{-2}. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the

(a) force on the floor by the crew and passengers,

(a) The normal force on the floor will be the reaction force due to crew and passengers.        Using Newton's laws of motion we can write :                                              Thus                                      or                                                  or                                                The direction of normal force on the floor will be vertically upward. 
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