Unit-1. Force & Motion. Solutions 1.3 Newton s Laws of Motion page - 40

Transcription

1 page Three blocks of masses 5 kg, 3 kg and 2 kg are tied with ropes and pulled by a horizontal force of F=60 N. The coefficient of friction between the blocks and the surface is 0,2. Take all the objects as a single body having mass of (5+3+2)=10 kg and draw a free body diagram to show all the forces on a diagram. y F N =100 N Calculate the acceleration of the blocks. (g=10 m/s 2.) F f =100.0,2 F f =20 N F=60 N x F net =m total.a (60-20)=10.a a=4 m/s 2 W=100 N

2 page - 40 y y 2. Two blocks are connected by a string on a frictionless inclined plane as shown in the figure. When the system is released, tension of the rope is measured to be 24 N. 60.sin30 o F N2 T 30 o 60.cos30 o x F N1 T 10m.sin30 o 30 o 10m.cos30 o x 10m 60 N Calculate the mass of m 1. (g=10 m/s 2. Pulleys are supposed to be weightless and frictionless.) 24-5m=m F net =m.a F net =m.a (60.0,5)-T=6.a T-(10m.0,5)=m.a 30-24=6.a a=1 m/s 2 m=4 kg

3 page - 40 y 3. The inclined planes given in the figure are all frictionless. The masses of the objects are related as m K =m L <m m and β is greater than α. F N mg.sina F net =m.a mg.sink=m.a mg.cosa k x a=g.sink mg Compare the accelerations of the objects when they are released. Acceleration depends on the gravitational acceleration and the angle of inclined plane. As the angle gets bigger acceleration gets bigger. It is independent on the mass. So; a L >a K =a M

4 page The surfaces given in the figures are all frictionless. The masses of objects X, Y and Z are m, 2m and 3m respectively. When the systems are released, the accelerations of the systems are a 1 and a 2. The tensions in the connecting ropes are T 1 and T 2. For Figure-I; mg=(m+2m).a 1 For Figure-II; mg=(m+3m).a 2 mg=3m.a 1 mg=4m.a 1 a 1 =g/3 a 2 =g/4 T 1 =2mg/3 T 2 =3mg/4 Calculate the ratio of accelerations and tensions. (Pulleys are supposed to be weightless and frictionless.) a 1 4 = a 2 3 and T 1 8 = T 2 9

5 page When the system given in Figure-I is released, object Y starts to move rightward with an acceleration of a. When the system given in Figure-II is pulled by a horizontal force of F, object Y starts to move leftward with an acceleration of a. For Figure-I; 2mg=(m+2m).a 2mg=3m.a a=2g/3 For Figure-II; F-2mg=(m+2m).a F-2mg=3ma F=3ma+2mg What is the magnitude of force F in terms of mg? (All the friction forces and the weight of the pulleys are ignored.) F=3m(2g/3)+2mg F=2mg+2mg=4mg

6 page The masses of the objects K, L and M are 2 kg, 1 kg and 2 kg respectively. The system given in the figure is released. After a while the rope connecting objects K and L breaks down. What can be said about the motion of object M? (All the friction forces and the weight of the pulley are ignored.) Firstly, the total weight on the right side is greater. K and L start to move downward and M starts to move upward with a constant acceleration. When the rope between K and L breaks down, the net force on the system will be zero. M will move at steady speed which is equal to the speed that it has while the rope breaks down.

7 page Object K is fired from the bottom of an inclined plane with an initial speed of v, it can reach point X then moves downward. The magnitude of its acceleration toward point X is a 1 and the magnitude of its acceleration when it moves downward is a 2. The coefficient of friction is 0,5. moving upward; F net =m.a (-mg.sin37 o )-(F f )=m.a 1 (-mg.sin37 o )-(0,5.mg.cos37 o )=m.a 1 (-6)-(4)=a 1 then a 1 =-10 m/s 2 moving dowward; F net =m.a (mg.sin37 o )-(F f )=m.a 2 (mg.sin37 o )-(0,5.mg.cos37 o )=m.a 2 What is the ratio of a 1 to a 2? (g=10 m/s 2.) (6)-(4)=a 2 then a 2 =2 m/s 2 a 1 = 5 a 2

8 page An object of mass m is stationary on a horizontal surface. Then a horizontal force of F is applied on the object. The acceleration of the object versus the magnitude of the applied force graph is given. Assume that the magnitude of the maximum static friction force is equal to the magnitude of the kinetic friction force. F f =12 N F net =m.a F-F f =m.a 27-12=m.5 m=3 kg F f = µ.f N Calculate the mass of the object and the coefficient of friction. 12= µ.3.10 then µ=0,4

9 page - 42 y F f 9. The coefficient of friction between the car and the box is 0,5. F fictitious =ma F N x mg What would be the minimum magnitude of the acceleration of the system in order to keep the box stationary with respect to the car? (g=10 m/s 2.) F f = µ.f N F fictitious = ma Object is stationary; F fictitious = F N and F f =mg µ.m.a =m.g 0,5.m.a=m.10 a=20 m/s 2

10 page - 42 y F N 10. The inclined plane moves with a constant acceleration. The box of mass m is stationary with respect to the inclined plane. ma ma.sinα mg.cosα ma.cos ma.sin ma.cosα α nα mg mg.sinα mg.cos x Express the acceleration of the inclined plane in terms of inclination angle α. (Ignore friction.) Object is stationary; m.g.sinα=m.a.cosα a=g.tanα

11 page The elevator moves upward with a constant acceleration of 3 m/s 2. T mg ma object is in equilibrium; What is the magnitude of the tension in the rope that connects the 2 kg object to the ceiling of the elevator? (g=10 m/s 2.) T=mg+ma T=(2.10)+(2.3) T=26 N

12 page - 42 y 12. The horizontal surface is frictionless and the coefficient of friction between 1 kg object and 4 kg object is 0,3. F fictitious =ma=1.a=a F N =10 N F f =0,3.10=3 N x What is the maximum magnitude of the horizontal force F in order to keep the 1 kg object stationary with respect to 4 kg object? (g=10 m/s 2.) F fictitious =F f a=3 m/s 2 10 N They move together. F=(1+4).3=15 N

13 page The elevator moves upward with a constant acceleration of 2 m/s 2. a) T T motion 2a=4 N motion 3a=6 N 20N 30 N T-24=2.a / 36-T=3.a / a) What is the magnitude of the acceleration of object X with respect to the elevator? b) What are the magnitudes of the accelerations of objects X and Y with respect to the ground? (All the friction forces and the weight of the pulley are ignored and g=10 m/s 2.) 36-24=5.a / a / =2,4 m/s 2 b) a X =0,4 m/s 2 (downward) a y =4,4 m/s 2 (upward)

14 page - 43 y 14. A block of mass 1 kg is placed on an inclined plane of mass 3 kg. Inclined plane is pushed by a horizontal force of 40 N as shown in the figure. ma 0,8.mg 37 o 0,6.ma 0,8.ma 37 o F N 0,6.mg x F=m.a system 40=4.a system a system =10 m/s 2 mg Calculate the acceleration of the block with respect to the inclined plane. (Ignore friction and g=10 m/s 2.) 0,8.ma - 0,6.mg = m.a 8-6 = 1.a a = 2 m/s 2 (upward)

15 page When the car is accelerating toward right with a constant acceleration of a, boxes A and B remain stationary with respect to the car. What is the magnitude of the acceleration of the car (a)? (All the friction forces and the weight of the pulley are ignored and g=10 m/s 2.) For object B; It is stationary with respect to the car. 20-T=0 T=20 N For object A; It is stationary with respect to the car. T=m A.a system 20=4.a system a system =5 m/s 2

16 page The horizontal surface is frictionless and the coefficient of friction between 2 kg object and 3 kg object is µ. When the system given in the figure is released, all objects move together with maximum acceleration that they can move together. For the system of the objects; F net =m total.a system 5.10=(2+3+5). a system a system =5 m/s 2 For 2 kg object; Calculate the coefficient of friction (µ). (Pulley is frictionless and its weight is ignored. g=10 m/s 2.) F fictitious =F friction then m. a system =µ.m.g µ= (a system /g)=(5)/(10)=0,5