Unit 2: Forces Trigonometry In this unit we are going to use a mathematical tool that is called trigonometry. Trigonometry expresses the relationship between angles and lengths in right angled triangles. You need to learn what sine, cosine and tangent of an angle are: For example: If you know the sine, the cosine or the tangent of an angle, you can calculate this angle using the arcsine, the arccosine or the arctangent. For example, in this case: (you could have also used the arccosine or the arctangent) Forces Forces can make an object change its motion, its direction or even its shape. Forces are vectors. Not only is it important to know the magnitude of the force, but it is also important to know where the force is applied and its direction. We are going to express the forces using the unit vectors and. The unit of forces in the international system is the Newton. 1
For example: We can calculate the length of this vector using the Pithagorean theorem: Weight It is the force of attraction between the Earth (or any other planet or moon) and an object located on it. It is calculated with this formula: Where W is the weight expressed in N, m is the mass expressed in kg and g is the acceleration of gravity of the planet, expressed in m s -2. The acceleration of gravity on the Earth is 9.8 m s -2. Normal force When we stand on a horizontal surface, why don t we sink in it, if there is a force (the weight) that is directed to the center of the Earth? Because there is another force, called the normal force. The addition of these two forces is 0 and that s why we don t sink. The normal force is always perpendicular to the surface. 2
Force of friction It is the force exerted by a surface on an object when it is moving (or trying to move) across it. It opposes the movement of the body. It can be calculated using the formula: Where is the coefficient of friction (it has no units) and N is the normal force. The sliding friction (when the body is moving) is slightly smaller than the static friction (when the body is not moving), because the coefficient of friction of the sliding friction is slightly smaller than the coefficient of friction of the static friction. That is why it is easier to move an object once it is already moving. Newton s laws Newton s first law of motion: every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. https://www.youtube.com/watch?v=q0wz5p0jdeu Newton s second law of motion: the relationship between an object's mass m, its acceleration, and the applied force is. https://www.youtube.com/watch?v=wzvhuq5rwje Newton s third law of motion: for every action there is an equal and opposite reaction. https://www.youtube.com/watch?v=cp0bb3wxj_k Centripetal force When we have a uniform circular motion, the velocity changes although the speed remains constant (because the direction of the velocity changes, remember that velocity is speed with a direction). Whenever there is an acceleration, there is a force, and in this case this force is called centripetal force. Remember that the centripetal acceleration is: Therefore, the centripetal force is: a c =v 2 /R F c =m a c =m v 2 /R This force is directed to the centre of the circumference. 3
Hooke s law We are going to use this law to calculate the force exerted by a compressed or extended spring on an object that is attached to it. The length of the compression or extension of the spring is proportional to the force the object suffers. The relationship between the length and the force is: Where F is the force expressed in N, k is a constant whose value depends on the spring and x is the length of the compression or extension. If k is expressed in N/m, x must be expressed in m. If k is expressed in N/cm, x must be expressed in cm The (-) on the formula means that the force of the spring and the compression or extension always goes in opposite directions. We can also use this law to calculate the force needed to compress or stretch a spring a certain distance. In this case the formula is: 4
Exercises 1. Draw on a cartesian coordinate system and calculate the length and the angle they form with the horizontal axis for the following forces: 2. One force whose length is 40 N has a direction North-East and forms an angle of 30º with the horizontal axis. Express it as a vector using the unit vectors and. 3. One force whose length is 20 N has a direction South-East and forms an angle of 70º with the horizontal axis. Express it as a vector using the unit vectors and. 4. One force whose length is 65 N has a direction North-West and forms an angle of 45º with the horizontal axis. Express it as a vector using the unit vectors and. 5. One force whose length is 27 N has a direction South-West and forms an angle of 60º with the horizontal axis. Express it as a vector using the unit vectors and. 6. Three forces are applied on one body. These forces are, and. What is the total force applied on the body? Calculate its length and draw it on a Cartesian coordinate system. 7. Four forces are applied on one body. These forces are,, and. What is the total force applied on the body? Calculate its length and draw it on a Cartesian coordinate system. 8. What is the total force applied on the box? 9. What is the total force applied on the box? 5
10. Calculate the total force. Mass, weight and normal force 11. Calculate the weight on the surface of the Earth of a man whose mass is 80 kg. Calculate his weight on the surface of the Moon. (g Moon =1.6 m s -2 ). 12. Calculate the normal force in the following cases, drawing a free-body diagram: a) A box of 2.5 kg is at rest on a horizontal surface. b) A box of 2.5 kg is at rest on a horizontal surface and is pushed down by a force of 8 N. c) A box of 2.5 kg is at rest on a horizontal surface and is pulled up by a force of 8 N. 13. Calculate the acceleration of gravity in Jupiter if an astronaut whose mass is 70 kg weighs 1820 N there. Newton s second law Remember: in LUAM 14. A force of 64.8 N is exerted upon a body of 12 kg of mass, initially at rest. Calculate: a) The acceleration acquired by the body. b) The velocity it will have in 2.5 seconds. c) The space it will have travelled in 2.5 seconds. 15. a) Calculate the force you need to apply upon a sledge of 4.5 kg of mass so that it acquires an acceleration of 8 m/s 2. b) Calculate its velocity after 3.5 seconds if it was originally at rest. c) Calculate the space it has travelled in those 3.5 seconds. 6
16. The table shows the forces applied to a body and the accelerations it acquires in each case: F (N) 3 6 9 12 15 a (m/s 2 ) 8 16 24 32 40 a) Draw a graph representing the force versus the acceleration using the values of the table. b) Calculate the mass of this body. 17. A body of 15 kg of mass moves at a constant speed of 36 km/h. Calculate the force you need to apply during 5 seconds so that it stops. 18. A force is applied upon a body of 3 kg of mass and makes its velocity rise from 1 m/s to 5 m/s in 3 seconds. Calculate the value of the force. 19. Two men apply a force upon a big box of 10 kg. These forces are of 4 N and of 9 N. Calculate: a) The acceleration that the box acquires. b) The velocity it has after 10 seconds, if the box was originally at rest. c) The space it will have travelled after those 10 seconds. in these two situations: i) The forces are applied in the same direction. ii ) The forces are applied in opposite directions. 20. A ball is kicked with a force of 275 N. The impact of the foot with it lasts 0.1 seconds and thanks to this impact the ball acquires a speed of 20 m s -1. What is the mass of the ball? 7
Force of friction On a horizontal surface a box of 20 kg is pulled by a horizontal force of 30 N. The coefficient of friction of the surface is 0.1. a) Calculate the acceleration the box acquires. b) Calculate the space it travels in 5 seconds and the speed after those 5 seconds, if the box was originally at rest. a) The box will move on the X axis but it will not move on the Y axis, what means that the addition of all the forces on the Y axis is 0. Y axis: N-m g=0 N=m g=20 9.8=196 N X axis: F fr = N=0.1 196=19.6 N F-F fr =m a b) v=v 0 +a t=0+0.52 5=2.6 m/s ----------------------------------------------------- 21. A wardrobe is pushed with a horizontal force of 580 N. If the coefficient of friction is 0.4, calculate: a) The acceleration it acquires (the wardrobe has a mass of 60 kg). b) The velocity and the distance travelled in 5 seconds (the wardrobe was initially at rest). 22. A sailing boat of 200 kg is pushed by the air with a force of 300 N. At the same time, there is a force of friction with the water of 100 N. a) What is the acceleration? b) What is its speed after 20 seconds if it was initially at rest? 23. A body of 2 kg which moves with a speed of 20 m/s reaches a rough surface and after 5 seconds it stops. Calculate the coefficient of friction of this surface. 8
24. A body of 4 kg of mass is at rest along a horizontal surface. When you apply a force of 20 N, it acquires an acceleration of 1 m/s 2. Calculate: a) The friction force. b) The coefficient of friction. c) The acceleration it would acquire if there wasn t any friction. 25. A body whose mass is 7 kg moves with a speed of 9.2 m/s on a horizontal surface. A man applies a force of 3 N in the opposite direction of the motion, because he wants to stop the body. The coefficient of friction of the floor is 0.15. Calculate: a) The acceleration the body acquires. b) The time it takes the body to stop. c) The space the body will have travelled before it stops. ------------------------------------------------------------------ On a horizontal surface a box of 20 kg is pulled by a force of 30 N that forms an angle of 30º with the floor. The coefficient of friction of the floor is 0.1. a) Calculate the acceleration the box acquires. b) Calculate the space the box travels in 5 seconds and the speed after those 5 seconds. a) F x =F cos30=30 cos30º=25.98 N F y =F sen30=30 sen30º=15 N Y axis N+F y -m g=0 N=m g-f y =20 9.8-15=181 N X axis F fr = N=0.1 181=18.1 N F x -F fr =m a b) v=v 0 +a t=0+0.394 5=1.97 m/s ----------------------------------------------------------------------------- 9
26. A man pulls a box of 20 kg with a force of 23 N that forms an angle of 10º with the horizontal floor. Calculate the acceleration the box acquires, knowing that the coefficient of friction is 0.1. 27. A box is pulled by a force of 30 N that forms an angle of 30º with the horizontal floor. Calculate the acceleration the box acquires, knowing that the coefficient of friction is 0.15. The mass of the box is 15 kg. 28. Antonio pulls a box of 40 kg with a force of 85 N that forms an angle of 20º with the horizontal floor. At the same time Sandra pushes the box with a horizontal force of 70 N. Calculate the acceleration the box acquires, knowing that the coefficient of friction is 0.2. What will be the space the box will have travelled in 7 seconds? 29. You are dragging a box at a constant velocity on a horizontal surface whose coefficient of friction is 0.1. The box has a mass of 10 kg. Calculate: Ropes a) The force of friction of the surface. b) The force that you are applying. 30. Two boxes are hanging by a rope on both sides of a pulley, as you can see on the figure: One of the boxes has a mass of 11 kg and the other box has a mass of 4 kg. Calculate the acceleration of the motion and the tension of the rope. 10
31. Calculate the acceleration of the motion and the tension of the rope for the following situation: The box of the table has a mass of 10 kg and the other box has a mass of 20 kg. 32. Repeat the 31 st problem considering that the coefficient of friction of the table is 0.1. 11
Inclined plane On an inclined plane that forms an angle of 35º with the floor, a box of 20 kg is pulled with a force of 140 N that is parallel to the surface. a) If the coefficient of friction is 0.1, calculate the acceleration the box acquires. b) If the end of the inclined plan is at a height of 60 cm, calculate the time the box will need to reach the end of the inclined plane, if it was originally at rest. a) Axis that is perpendicular to the plane (Y axis) N-W y =0 N=W y =W cos =m g cos Axis that is parallel to the plane (X axis) W x = W sen =m g sen F fr = N= m g cos F-F fr -W x =m a 0.58 b) =1.9 s -------------------------------------------------------------- 12
33. Calculate the acceleration of a box of 20 kg of mass that falls on an inclined plane that forms an angle of 30º with the horizontal plane. The coefficient of friction with the plane is 0.1. 34. Calculate the acceleration of a box of 40 kg of mass that falls on an inclined plane that forms an angle of 60º with the horizontal plane. The coefficient of friction is 0.15. Centripetal Force 35. Calculate the acceleration and the centripetal force that a body has if it has a mass of 1500 kg and rounds a curve of 20 m of radius at a constant speed of 80 km/h. 36. A body makes a uniform circular motion with an angular speed of 3 rpm (revolutions per minute). The radius of the circumference is 4 metres. Calculate: a) The angular speed in rad/s and the linear speed in m/s. b) If the body has a mass of 5 kg, calculate the centripetal force of the motion. 37. On a circular motion whose radius is 2 metres and with constant angular speed, a body of 1.5 kg of mass travels an angle of 15 rad in 3 seconds. Calculate: a) The angular speed in rad/s. b) The angular speed in rpm. c) The linear speed in m/s and in km/h. d) The angle and the space the body has traveled in 2 hours and a half. e) The centripetal force. 13
Hooke s law 38. When we hang a body of 3.5 kg from a spring it extends 19 cm. Calculate the constant of the spring. 39. A spring has a constant of 120 N/m. Calculate how much it will extend when a body of 6 kg of mass is hung. 40. What is the mass of the body that must be hung on a spring whose constant is 9 N/cm for it to extend 50 mm? 14