NEWTONS 1st LAW 1. A 10.00 kg mass is tied to a string with a maximum strength of 100 N. A second string of equal strength is tied to the bottom of the mass. a) If the bottom string is pulled with a jerk the bottom string breaks. Why? b) If the bottom string is pulled slowly the top string breaks. Why? 2. Why does a child on a toboggan fall backward when the toboggan is given a sharp tug? 3. In a recent motor accident two cars collided head-on while traveling in opposite directions at 40 km/h. A passenger in the front seat of one of the cars was not wearing a seat belt. She was injured and died in hospital three hours later. Explain this tragedy in terms of Newton's first law of motion. 4. Explain the following by reference to Newton's first law: a)you stamp your feet to get the snow off your boots. b) A railway worker has to take care when jumping from the side of a moving railcar. 5. What does Newton's first law tell us about the motion of the ball that the astronaut has thrown? 6. When the ring is given a sharp grab, the chalk balanced on top will fall directly into the bottle. Completely explain the why this happens. Use the appropriate terminology. 7. Explain why the student survives the blows of his slightly demented instructor. NEWTONS 2ND LAW 1. Fill in the following blanks. a) When acted on by balanced forces an object will experience motion but when acted on by an unbalanced force it will experience. b) The acceleration of a falling mass due to gravity is m/s 2. c) The distance travelled per unit of time is an object's. d) The rate of change of speed with time is an object's. e) If an object is travelling at 50 m/s, and the forces acting on it are completely balanced, in two seconds its velocity will be. f) An object which moves more slowly each successive second towards the east has, acceleration. 2. A 5.00 kg mass is falling through the water at a constant 4.00 m/s. What is the buoyant force on the mass? [49.1 N] 3. What net force acts on a 5.0 kg object if it is being pushed with a force of 50 N from the east and 35 N from the west when the frictional force is 10 N? [5 N] 4. A force of 5000 N is pulling a 400 kg mass upwards. What is the net force on the mass? [1076 N] 5. A given mass acquires an acceleration of 2.00 m/s 2 when acted upon by a force of 8.00 N. What is the mass of the object? [4.00 kg] 6. What net force is required to accelerate a 50 g object At 40 cm/s 2? [2.0 x 10-2 N] 7. If the force of gravity on a body is 12.00 N on the surface of the moon where g = 1.62 m/s 2 what is its mass? [7.41 kg] 8. A given net force acts on a 2.00 kg mass and gives it an acceleration of 3.00 m/s 2. What acceleration is produced by the same force when acting on a mass of 4.00 kg? [1.50 m/s 2 ] 9. A 850 kg car is drawn vertically up a mine shaft. In starting the car the acceleration is 5.45 m/s 2. Determine the applied upwards force acting on the car during this period. [1.297 x 10 4 N] 10. A chord passes over the top of a pulley with a 7.25 kg mass attached to one end and a 8.75 kg mass attached to the other end. Determine the acceleration of the blocks. Neglect any friction [0.920 m/s 2 ] 1
11. A chord passes over the top of a pulley with a 10.36 kg mass attached to one end and a 7.75 kg mass attached to the other end. Determine the acceleration of the blocks. [ 1.41 m/s 2 ] 12. A chord passing over a frictionless pulley has a 6.5 kg mass tied on one end and a 8.75 kg mass tied on the other end. Compute the acceleration of the system. [1.4 m/s 2 ] 13. A 50 000 kg rocket in space fires its engines for 60.0 s. The thrust of the engines is 200 000. N. What is the final speed of the rocket if the initial speed of the rocket 830 m/s? [1.07 x 10 3 m/s] 14. A 4.0 x 10 5 kg space platform must be brought from a speed of 2.2 x 10 3 m/s to 2.5 x 10 3 m/s. It will do this by firing a booster rocket with a thrust of 1.0 x 10 5 N. How many seconds must the rocket burn in order to achieve this speed? [1.2 x 10 3 s or 20 min] 15. A weather balloon and its instruments have a combined mass of 127 kg. The balloon is released on a calm day and has an upward lift of 1500 N. After accelerating, from rest, for 80.0 s the instruments are turned on to sample the atmosphere. What height will the balloon then be at? [6.40 x 10 3 m] FRICTION 1. Find the normal force in each of the following situations involving a 5.00 kg mass: a) the mass is sitting on a level floor. [49.4 N] b) the mass is on a 30 o incline. [42.5 N] c) the mass is pushed against a vertical wall with a force of 20.0 N. [20 N] 2. If the coefficient of static friction is 0.400 in each of the cases above, how much frictional force is present in each case? [19.62 N, 17.0 N, 8.00 N] 3.A 2 500 kg car is sliding to a stop on asphalt (µ k = 0.550). What is the force of friction on the car? [1.35 x 10 4 N] 4. A 250 g eraser is pushed against a whiteboard with a force of 30.0 N. If the coefficient of kinetic friction is 0.220, how much frictional force is involved in erasing the board? [6.60 N] 5. a) Draw in the triangle representing the weight of the shoe, F g, the parallel force, F //, and the perpendicular force, F per. b) Explain why the shoe starts to slide down the incline as the angle is increased even though no push is given to the shoe. 6. Define the following: a) force of friction: c) limiting static force of friction: b) normal force: d) kinetic force of friction: 7. A toboggan loaded with camping equipment has a mass of 15 kg and rests on a snow-covered lake. The coefficient of static friction of the snow is 0.400 and its coefficient of sliding friction is 0.200. You have been given the job of pulling this toboggan over the lake. a. Determine the force needed to start the toboggan moving. [58.9 N) b. Determine the force needed to keep the toboggan moving at a constant velocity using a friction equation. [29.4 N] 8. The combined mass of you and your skateboard is 50.0 kg. Someone pulls on the skateboard so that it accelerates at 2.0 m/s 2. If the force of friction between the skateboard and the road is known to be 30.0 N what is the size of the pulling force? [1.3 x 10 2 N] 9. The combined mass of you and your bicycle is 120.0 kg. If you pedal with a force of 400.0 N and if the coefficient of friction is 0.250 N, what acceleration will you undergo? [0.881m/s 2 ] 10. What will be the acceleration experienced by a 45.0 kg mass if a 45.0 N force is applied and the frictional forces present are 27.0 N? [0.400 m/s 2 ] 11. The combined mass of you and your skateboard is 50.0 kg. If you start at rest, and someone pulls on you with a force of 40.0 N and if the force of friction is 25 N, how long will it take you to cover a distance of 35 m? [ 15 s] 12. There is a constant frictional force of 120 N between your skateboard and a level road. You and the skateboard have a total mass of 60.0 kg. If you have an initial velocity of 10.0 m/s how far will you be able to roll on level ground? [25.0 m] 2
13. A 850 kg car skidding on an icy highway at 32.5 km/h slides onto some bare pavement. If the coefficient of friction between the tires and the bare pavement is 0.200, what time(s) does it take for the car to stop? [4.60s] 14. During a heavy rainfall, a student driving his car at 90.0 km/h on a highway must suddenly slam on the brakes. The brakes cause all four wheels and tires to lock and the car skids. If the student and car have a mass of 785 kg and the coefficient of sliding friction between the tires and the highway is 0.500, how far will the car slide? [63.7 m] 15. A student attempts to slide a 40 N wooden block that is resting on a lab table. He exerts a pulling force of 10 N but the block does not move so he increases the force. The block begins to slide when the pulling force reaches 20 N. The block continues to move at a constant velocity when the pulling force is reduced to 15 N. a. Calculate the maximum force of friction while the block was at rest on the table. [~20 N] b. Calculate the coefficient of friction [static) while the block was at rest on the table. [0.50] c. Calculate the coefficient of friction (kinetic) while the block was sliding on the table. [0.38] 16. A trunk weighing 562 N is at rest on a plane inclined at 30.0 to the horizontal. a. Find the force parallel. [281N] b. Find the force perpendicular [487 N] c. Which of the forces acts to move the trunk down the incline plane? d. Which of the forces acts to hold the trunk on the incline plane? 17. The same trunk is now resting on an plane inclined at an angle of 50.0 to the horizontal a. Find the force parallel [431 N] b. Find the force perpendicular [361N] c. Assuming the coefficient of friction is 0.200, what is the acceleration of the trunk down the incline? 18. A car with the mass of 1.2 x 10 3 kg is parked on a slope of 36.0. a. Find the force tending to cause the car to roll down the hill. [7.0 x 10 3 N] b. What is the magnitude of the force parallel? c. If the coefficient of friction is µ k = 0.225, what is the acceleration of the car down the hill? d. What distance will the car cover in 1.80 s? 20. A 16 kg box rests on a incline plane at an angle of 52. If the coefficient of friction is 0.12 what is the acceleration of the box? [7.0 m/s 2 ] 21. Use the diagram showing a block moving down an inclined plane to answer the question The coefficient of sliding friction between the 10.0 kg block and the plane is 0.35. Find the acceleration of the block down the plane. [3.2 m/s 2 ] 22. A 87 kg block slides down a 31 slope as shown in the diagram. The coefficient of friction between the block and the surface is 0.25. What distance doe the block move in 1.20 s? 23. A 1200 kg sports car is accelerating up an 18 hill at 0.055 m/s 2. The coefficient of static friction between the wheels and the road is µ k = 0.88. What force is the engine exerting? [1.4 x 10 4 N] 24. Use the diagram showing a 4.50 kg block starting from rest and moving down an inclined plane to answer the question. The coefficient of sliding friction between the block and the plane is 0.205. Find the distance the block travels down the plane in 0.500 s. [0.73 m] 19. What is the frictional force if a 5.00 kg block remains stationary on the incline plane with an angle of inclination of 15? [12.7 N] 3
25. An 18 kg cart is connected to a 12 kg hanging block as shown. (Ignore friction.) a) Draw and label a free body diagram for the 18 kg cart. b) What is the magnitude of the acceleration of the cart? [7.3 m/s 2 ] c) How fast will the 18 kg cart be moving after 1.5 s? [11 m/s] NEWTON'S 3RD LAW 1. The horse claims that if he pulls on the cart, the cart will simply pull back on him with an equal and opposite force and so there is no way that he can move the cart. What is the flaw in the horse's argument? 26. A 150 kg crate starts from rest and slides down a 25.0º incline plane with uniform acceleration. The length of the incline is 8.00 m. The crate reaches a velocity of 5.00 m/s at the bottom of the incline. Find the coefficient of friction between the crate and the plane. [0.291] 27. A block M of mass 3.0 kg is placed 2.5 m from the point X on a frictionless plane as shown below. The plane is inclined at an angle of 45. 2. What is the expected result concerning the relative velocities of the orange and the apple if the tension in the rope is 5.00 N. Assume that they have the same mass of 1.00 kg, and that µ s is 0.880 for the apple and 0.150 for the orange 3. Explain the action-reaction principle which causes a car to move forward. Calculate: a) the force acting on M down the plane [21 N] b) the acceleration of M. [6.9 m/s 2 ] c) the velocity of M at point X, assuming that it starts from rest. [5.9 m] d) M is once again placed on the inclined plane. A second block, m, of mass 1.5 kg is now attached to it using a frictionless pulley and a massless string as shown. Calculate the time it will take mass M to travel 0.575 m. [0.92 s] 4. Explain how the two masses could give the same tension in the rope as a single mass does. 5. Two men involved in a tug-o-war are pulling with equal forces of 500 N. The ropes are connected by an accurate spring balance which measures force in Newtons. Explain the reading on the balance? 4
6. a) What force is required to hold a 2400 kg helicopter in a stationary hovering position? [2.35 x 10 4 N] b) Name the force pairs that relate to the Second Law and the force pairs that relate to the Third Law. c) Are the forces in both pairs equal in magnitude? 13. Does the device work? Explain your answer. 7. Explain why the magnetic propulsion car show does not move. 8. Find the 'apparent weight' of a 75.0 kg man in each of the following situations: a) moving upwards at a constant rate of 4.5 m/s b) accelerating upwards at 2.00 m/s 2 c) decelerating upwards at 1.50 m/s 2 d) accelerating downwards at 2.50 m/s 2 [a) 736 N b) 886 N c) 623 N d) 548 N] 9. A 3200 kg mine car starts down a vertical mine shaft with an acceleration of 1.21 m/s 2. Determine the applied upwards force on the cable at the start of the descent. [2.75 x 10 4 N] 10. A 50.0 kg bale hangs at the end of a rope. Find the acceleration of the bale if the upwards force acting on the rope is; a) 490.5 N b) 735 N c) 392 N [a) 0 m/s 2 b) 4.90 m/s 2 c) -1.96 m/s 2 ] 11. Determine the force that a 75 kg man exerts on the floor of an elevator when a) it is at rest [740 N] b) rises with constant velocity of 2.5 m/s [ 740 N] c) rises with a constant acceleration of 2.5 m/s 2 [920 N] 12. Suppose you are stranded in the middle of a lake in a sailboat because the wind has died down. You have with you an electric fan, a battery to operate friends with lots of advice. One friend connect the fan to the battery and point it suggests you take down the sail and point rear of the boat. Whose advice is best? VECTORS IN 2 - DIMENSIONS 1. Add the following vectors together graphically to find the net vector. Add them in any order but maintain the same direction and magnitude. Copy the first vector on a quadrant system, through loose leaf sheet and then copy the second vector at the tip of the first and the third vector at the tip of the second. The net vector will go from the origin to the tip of the third vector. a) b) c) 2. Add the following vectors graphically to solve the problem. A man walks two blocks south and four blocks east. How far has he walked? How far is he from his starting point? [4.47 blocks 63 o E of S] 3. An ocean liner crosses the Atlantic, sailing eastward at 18 km/h. A man walks across the deck at right angles to the ship's keel at 3 km/h. Make a scale diagram to find the velocity of the man with respect to the water. [18 km/h @ 9.5 o N of E] 4. Add the following vectors graphically to solve the problem. On a wilderness survival test a team is told to travel 12.0 km east and 5.0 km north. What is the team's resultant displacement from the starting point? [13 km @ 23 o N of E] 5. A frisky dog runs 15.0 m to the right, then 25.0 m to the left. It then runs 8.0 m to the right again. How far did the dog run? What was the net displacement? [ 48.0 m, 2.0 m to the left] the fan, and several suggests you should at the sail. Another the fan to the Explain your answer. 5
6. A boat sails 45 km at a heading of 45 and then makes a turn and sails 35 km at a heading of 225. What is the distance traveled and the net displacement of the boat? [ 80 km, 10 km at 45 ] 16. Two soccer players kick the ball at exactly the same time. One player's foot exerts a force of 60.0 N north. The other's foot exerts a force of 80.0 N east. What is the magnitude and direction of the resultant force on the ball? [100 N of force at 53.1 E of N] 7. Draw scale diagrams to answer the following questions a) A man walks 18.0 km at 150.0 and then 25.00 km at 80.0 W of N. What is his net displacement? b) An airplane is headed 42.0 N of W at 360. km/h A wind from the south comes up with a velocity of 50.0 km/h. What is the plane's new velocity? 8. Draw a scale diagram to solve the following problem. A plane flying at 800.km/h at 24.2 o S of W, experiences a wind of 50.0 km/h blowing 22.0 o S of E. What is the resultant velocity of the plane? 9. A car travels 35.0 km E then 20.0 km W. What is the distance covered and the final displacement? [55 km, 15 km E] 10. What is the resultant displacement [both size and direction) if an airplane is flown 60 km south, then 60 km east? [84.85 km at 45 o E of S] 11. An airplane is headed due east with velocity of 500 km/h. The wind from the south has a velocity of 120 km/h. What is the plane's velocity with respect to the ground? [514 km/h at 13.5 o N of E] 12. A boat travels at 10.0 m/s. It heads straight across a river which flows at 3.00 m/s. What is the resultant velocity of the boat across the river? [10.4 m/s at 16.7 to the original direction of motion] 13. A plane flying due north at 100.0 m/s is blown due west at 50.0 m/s by a strong wind. Find the plane's resultant speed. [112 m/s at 26.6 W of N] 14. A hiker leaves camp and walks 10.0 km due north. The hiker then walks 10.0 km due east. a) What is the total distance walked by the hiker? [20.0 km ] b) Determine the total displacement from the starting point. [14.1 km at 45 E of N] 17. A canoeist paddles north across a river at 3.0 m/s. (The canoe is always pointed at right angles to the river current). The river is flowing east at 4.0 m/s and is 100. m wide. a) What is the velocity of the canoe relative to the river bank? [5.0 m/s 37 o to the river bank] b) Calculate the time required to cross the river.[33 s] c) How far downstream is the landing point from the starting point? [1.3 x 10 2 m] 18. A boy swims at 2.80 km/h heading west directly across a river which has a current flowing north at 6.5 km/h. What is the boy's velocity with respect to the shore? [7.1 km/h at 23 to the shore] 19. The pilot of a light plane heads due north at an air speed of 400 km/h. A wind is blowing from the west at 60.0 km/h. a) What is the plane's velocity with respect to the ground? [405 km/h at 8.50 o E of N] b) How far off course would the plane be after 2.5 h, if the pilot had hoped to travel due north but had forgotten to check the wind velocity? [150 km east] 20. A motorboat heads due east at 16.0 m/s across a river that flows due south at 9.00 m/s. a) What is the resultant velocity [speed and direction) of the boat? [18.4 m/s at 29.4 S of E] b) If the river is 136 m wide, how long does it take the motorboat to reach the other side? [8.50 s] c) How far downstream is the boat when it reaches the other side of the river? [76.5 m] 21. A motorboat travels at 40.0 m/s. It heads straight across a river 320.0 m wide. a) If the water flows at the rate of 8.00 m/s what is the boat's velocity with respect to the shore? [40.8 m/s at 78.7 ] b) How long does it take the boat to reach the opposite shore? [8.00 s] 15. An airplane flies due west at 120.0 km/h. At the same time the wind blows it due north at 40.0 km/h. What is the plane's resultant velocity? [126 km/h at 18.4 N of W] 6
VECTORS IN 2 - DIMENSIONS & COMPONENTS 1. Change the following rectangular coordinates into bearing coordinates with respect to north. Draw a diagram of the vector in each case. a) (-125, 39) m/s b) (-45, -60) N c) (4.0, -5.3) m/s 2 d) (900, 250) m a) [1.3 x 10 2, -73 o ] m/s b) [75, -1.4 x 10 2o ] N c) [6.64, 1.4 x 10 2o ] m/s 2 d) [934, 74.5 o ] m 2. Change the following bearing vectors into rectangular components. Draw a sketch of the vector in each case show and label the bearing angle. a) [40.0, 23.0 o ] m/s b) [400.0, -70.0 o ] N c) [68.0, 160.0 o ] km d) [150, 2.40 x 10 2o ] m/s a) (15.6, 36.8) b) (-376, 137) c) (23.3, -63.9) d) (-130, -75.0) 3. A micro-light airplane has a heading of 35 N of E and is traveling at 40.0 m/s. The wind begins to blow at 30.0 m/s in the direction 25 W of N. What is the resultant velocity of the airplane? [54 m/s at 68 N of E.] 4. An airplane headed 10.0 E of N at 200.0 m/s enters a jet stream blowing 65.0 E of S at 120 m/s. If the pilot does not correct his heading, what will be the velocity of the airplane? [205 m/s at 45.5 N of E] 5. A river flows at 4.00 m/s towards a direction of 20.0 W of N. A canoe heads 50.0 W of S at 2.00 m/s relative to the water. What will be the track velocity of the canoe in bearing form? [[3.81, 310. ]m/s] 8. Add the following vectors. In each case make a small sketch and show all your work for part marks. a) 5.8 m/s north and 6.3 m/s west b) [19.0, 85.0 ) km and [50.0, 180. ) km [8.6 m/s @ 43 o N of W] [51.9 km @ 159 o ] 9. Add the following vectors. Make a small sketch and show all your work for part marks. a) 7.8 m/s north and 4.8 m/s west b) [19.0 137 ] km and [12.0 85.0 ] km [9.2 m/s @ 58 o N of W. 28.0 km @ 117 o ] 10. The heading of a boat is [4.0, -115 ] m/s on a river having a current of [3.0, 40.0 ] m/s. What is the net velocity of the boat? [1.8, -70 o ] 11. A boat, heading 25.0 W of N at 32.0 km/h encounters a tidal current flowing at 12.0 km/h and 45.0 N of E. a) If no adjustment is made what is the track velocity of the boat? [37.8 km/h @ 82.3 o N of W] b) If the boat is to maintain it's original direction what heading should it take? [30.1 km/h @ 43.0 o N of W] 12. A ferry is headed at 15.0 S of W with a speed of 8.50 m/s. A tidal current of 4.60 m/s flows 24.0 W of N a) Find the track velocity of the ferry. [10.3 m/s @ 11.2 o N of W] b) To maintain the same direction and speed as before how should the ferry be directed? [9.01 m/s @45.3 o S of W] 13. A 750 N weight is supported by two ropes fastened together by a knot, as shown in the diagram. 6. What is net force on an object if a 20.0 N force acts on it 40.0 W of N and a 25.0 N force acts 30.0 W of S? [26.1 N at 14.0 S of W.] 7. Add the following vectors with results in bearing format. a) [13.0, 135 ] m/s and [12.0, 25.0 ] m/s b) [45, 340.0 ] m/s and [220.0, 195 ] km/h c) [15.0, 37 ] m and [45, -47 ] m d) [22.5, 225 ] km and [150.0, 45.0 ] km a) [14.4, 83.3 ]m/s b) [35, 240 ]m/s c) [49, -29 ]m d) [128, 45.0 ] km a) Draw a free-body diagram showing the forces acting on the knot. b) What is the tension in rope 2? [716 N] 14. Determine the scale reading in the right arm of the suspension The scale was set to read zero with no suspended load. [16 N] 7
5. A bomber is approaching a target at 60.0 m/s and 500. m above the ground. How far in front of the target should the bombs be released in order that they will hit the target? [606 m) 15. A floodlight is suspended from two cables as shown below. The tension in the right cable is 140 N. 6. An airplane drops a bomb from an altitude of 650 m as it is moving horizontally with a velocity of 45.0 m/s. How far in front of a target should the airplane drop the bomb in order to hit the target? [518 m) 7. A cannon fires a cannonball with a speed of 100m/s at an angle of 20 above the horizontal. What are the horizontal and vertical components of the initial velocity of the cannonball? [94 m/s, 34 m/s) What is the tension in the left cable? [91.4 N] 16. A 43.8 kg sign is suspended by two wires, as the drawing shows. Find the tension in wire 1. [T 1 = 249 N] 8. A projectile is fired horizontally from the top of a cliff with an initial speed of 3.0 x 10 2 m/s and strikes the ground below after 9.4 s. How tall is the cliff? What is the projectile's range? [4.3 x 10 2 m, 2.8 x 10 3 m] 9. A plane travels 40.0 o north of east for a distance of 300.0 km. How far north and how far east does the plane travel? [230 km E, 193 km N] PROJECTILE MOTION 1. A man jumped horizontally from a window and reached the ground 0.80 s later. If he landed 3.6 m from the base of the building, what was his initial horizontal speed? How high was the window above the ground? [ 4.5 m/s, 3.1 m] 2. A bullet was fired horizontally from a rifle and struck the ground 1.5 s later after traveling 4.0 x 10 2 m. What was the muzzle speed of the bullet? From what height was it fired? [ 2.7 x 10 2 m/s, 11 m] 3. A steeplechase runner must jump from the top of a 2.0 m high fence and clear a water-filled ditch below which is 3.0 m wide. If she jumps horizontally, at what speed must she leave the fence? [4.7 m/s] 4. A cannon ball is fired horizontally from a fort with an initial velocity of 1.8 x 10 2 m/s and strikes a ship at sea below 7.2 s later. What is the ball's range? How high is the fort above sea level? [2.5 x 10 2 m] 10. An arrow is shot at 50.0 m/s and 25.0 above the horizontal. a) What are the horizontal and vertical components of the arrow's velocity? [45.3 m/s, 21.1 m/s] b) To what height does the arrow rise? [22.7 m] c) How far does the arrow travel horizontally? [195 m] 11. Brian is in a car moving at 80.0 km/h. He throws a water bomb at a sign 50.0 m away which is on the same level as the car. If he throws the water bomb directly upwards, with that velocity should he throw it so that it will hit the sign? [11.0 m/s] 12. A paper airplane is thrown into the air with a velocity of 7.5 m/s at an angle of 56.0 above the horizontal. If there is a 12.5 m/s tailwind blowing at the time, how far can the airplane be expected to travel horizontally? [21 m] 13. A placement kicker for a football team kicks a football at 23.0 m/s at 43 to the horizontal. If there is a 15.0 km/h tailwind in the first quarter what is the distance that he will be able to kick the ball? [67 m] 14. A ball is thrown into the air with a velocity of 10.5 m/s at an angle of 62.0 above the horizontal. There is a 8.50 m/s tailwind blowing at the time. How far can the ball be expected to travel horizontally? [25.4 m] 8