Laws of Motion

5.31 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 ?

5.30 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 ?

5.29 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 7th coin (counted from the bottom) due to all the coins on its top

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

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

5.28 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 ?

5.27 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

(a) Force on the floor by the crew and passengers

(b) Action of the rotor of the helicopter on the surrounding air

(c) Force on the helicopter due to the surrounding air

5.26 A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are : [Choose the correct alternative]

Lowest Point                          Highest Point

(a) mg – T₁                                mg + T₂

(b) mg + T₁                                mg – T₂

(c) mg +T₁   – (m ${\mathit{v}}_1^2$ ) / R  mg – T₂+ (m ${\mathit{v}}_1^2$ ) / R

(d) mg – T₁   – (m ${\mathit{v}}_1^2$ ) / R  mg + T₂ + (m ${\mathit{v}}_1^2$ ) / R

T₁ and v₁ denote the tension and speed at the lowest point. T₂ and v₂ denote corresponding values at the highest point.

5.25 Figure 5.18 shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with 1 m s^-2. What is the net force on the man? If the coefficient of static friction between the man’s shoes and the belt is 0.2, up to what acceleration of the belt can the man continue to be stationary relative to the belt ?(Mass of the man = 65 kg.)

5.24 Figure 5.17 shows the position-time graph of a body of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the body ? What is the magnitude of each impulse?

5.23 Explain why

(a) A horse cannot pull a cart and run in empty space

(b) Passengers are thrown forward from their seats when a speeding bus stops suddenly

(c) It is easier to pull a lawn mower than to push it

(d) A cricketer moves his hands backwards while holding a catch

5.22 If, in Exercise 5.21, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks :

(a) The stone moves radially outwards

(b) The stone flies off tangentially from the instant the string breaks

(c) The stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle ?