1. Forces

2. Unit 2: Forces Chapter 6: Systems in Motion
6.1 Motion in Two Dimension
6.2 Circular Motion
6.3 Centripetal Force, Gravitation, and Satellites
6.4 Center of Mass

3. 6.1 Investigation: Launch Angle and Range
Key Question:
Which launch angle will give a marble the best range?
Objectives:
Use the Marble Launcher to find the launch angle that produces the maximum range for a projectile.

4. Motion in two dimensions
Real objects do not move in straight lines alone; their motion includes turns and curves.
To describe a curve you need at least two dimensions (x and y).

5. Displacement Distance is scalar, but displacement is a vector.
A displacement vector shows a change in position.

6. Displacement
A displacement vector’s direction is often given using words.
Directional words include left, right, up, down, and compass directions.

7. Solving displacement problems
Displacement vectors can be added just like force vectors.
To add displacements graphically, draw them to scale with each subsequent vector drawn at the end of the previous vector.
The resultant vector represents the displacement for the entire trip.

8. Adding vectors Looking for: … the displacement.
A mouse walks 5 meters north and 12 meters west. Use a scaled drawing and a protractor to find the mouse’s displacement. Use the Pythagorean theorem to check your work.
Looking for: … the displacement.
Given: … distances and direction (5 m, N) and (12 m, W)
Relationships: Pythagorean theorem a2 + b2 = c2
Solution: Make a drawing with a scale of 1 cm = 2 meters. Measure the angle from the x axis.
Pythagorean theorem: (5)2 + (12)2 = c = c = c
The mouse walks 13 meters at 157°.

9. Velocity vectors
Velocity is speed with direction, so velocity is a vector.
As objects move in curved paths, their velocity vectors change because the direction of motion changes.
The symbol v is used to represent a velocity vector.

10. Velocity vectors
Suppose a ball is launched at 5 m/s at an angle of 37°
At the moment after launch, the velocity vector for the ball written as a magnitude‑angle pair is v = (5 m/s, 37°).

11. Velocity vectors
In x-y components, the same velocity vector is written as v = (4, 3) m/s.
Both representations tell you exactly how fast and in what direction the ball is moving at that moment.

12. Using velocity vectors
A train moves at a speed of 100 km/h heading east. What is its velocity vector in x-y form?
Looking for: … the velocity vector.
Given: … speed (100 km/h) and direction (east).
Relationships: x-velocity is east and y-velocity is north.
Solution: v = (100,0) km/h

13. Projectile motion
Any object moving through air and affected only by the force of gravity is called a projectile.
Flying objects such as airplanes and birds are not projectiles, because they are affected by forces generated from their own power.

14. Projectile motion
The path a projectile follows is called its trajectory.
The trajectory of a projectile is a special type of arch- or bowl-shaped curve called a parabola.

15. Trajectory and range
The range of a projectile is the horizontal distance it travels in the air before touching the ground.
A projectile’s range depends on the speed and angle at which it is launched.

16. Two dimensional motion
Projectile motion is two-dimensional because there is both horizontal and vertical motion.
Both speed and direction change as a projectile moves through the air.

17. A ball rolling off a table
The horizontal and vertical components of a projectile’s velocity are independent of each other.

18. Horizontal velocity
The ball’s horizontal motion looks exactly like the its motion if it was it rolling along the ground at 5 m/s.

19. Vertical velocity
The vertical (y) velocity increases due to the acceleration of gravity.

20. NOTE: These equations are suitable only for situations where the projectile
starts with zero vertical velocity, such as a ball rolling off a table.

21. Range of a Projectile
The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle.

22. Range of a Projectile
The range of a projectile is calculated from the horizontal velocity and the time of flight.
The air time and height are greatest when a ball is hit at an angle of 90°, but air time and height are zero when a ball is hit at an angle of 0°.

23. Projectile motion
A stunt driver steers a car off a cliff at a speed of 20.0 m/s. The car lands in a lake below 2.00 s later. Find the horizontal distance the car travels and the height of the cliff.
Looking for: … vertical and horizontal distances.
Given: … the time (2.00 s) and initial horizontal speed (20.0 m/s).
Relationships: Use : dx= vxt dy = 4.9t2
Solution: dx= (20 m/s)(2 s) = 40 m.
dy = (4.9 m/s2)(2 s)2 = (4.9 m/s2)(4 s2) = 19.6 m.

24. 6.2 Investigation: Launch Speed and Range
Key Question:
How does launch speed affect the range of a projectile?
Objectives:
Use the Marble Launcher to explore if a relationship exists between the launch speed and the range of a projectile.

25. Motion in Circles
We say an object rotates about its axis when the axis is part of the moving object.
A child revolves on a merry-go-round because he is external to the merry-go-round's axis. 25

26. Angular speed is the rate at which an object rotates or revolves.
There are two ways to measure angular speed
number of turns per unit of time (rotations/minute)
change in angle per unit of time (degrees/s or radians/s)
Motion in Circles 26

27. Angular speed
There are 360 degrees in a full rotation, so one rotation per minute is the same angular speed as 360 degrees per minute

29. Calculating angular speed
A merry-go-round makes 10 rotations in 2 minutes. What is its angular speed in rpm?
Looking for: … the angular speed in rotations per minute.
Given: … number of rotations (10) and the time (2 min.)
Relationships: Use: angular speed = rotation time
Solution: angular speed = 10 rotations = 5 rpm
2 minutes

30. The relationship between linear and angular speed
Each point on a rotating object has the same angular speed.
The linear speed of each child is not the same because they travel different distances.

31. The relationship between linear and angular speed
The distance traveled during one revolution equals the circumference of the circle.

32. The relationship between linear and angular speed
The linear speed (v) of a point at the edge of a turning circle is the circumference divided by the time it takes to make one full turn.
The linear speed of a point on a wheel depends on the radius, r, which is the distance from the center of rotation.

33. Calculating linear speed
The blades on a ceiling fan spin at 60 rpm. The fan has a radius of 0.5 m. Calculate the linear speed of a point at the outer edge of a blade in m/s.
Looking for: … the linear speed in m/s.
Given: … angular speed (60 rpm) and the radius (0.5 m)
Relationships: Use: v = 2  r t
Solution: The blades spin at 60 rotations per minute, so they make 60 rotations in 60 seconds. Therefore it takes 1 second to make one rotation. v = 2 (3.14) (0.5 m) = 3.14 m/s
(1 s)

34. Linear, rotational and rolling motion
Rolling is a combination of linear motion and rotational motion.
Holding a bicycle wheel up in the air and moving it to the right is linear motion.
If you lift a bicycle’s front wheel off the ground and make it spin, the spinning wheel is rotational motion.

35. Linear distance equals circumference
The distance the bicycle moves depends on the wheel’s size and angular speed.
When the wheel makes one full rotation, the bicycle goes forward one circumference of the wheel.

36. 6.3 Investigation: Levers and Rotational Equilibrium
Key Question:
How do levers work?
Objectives:
Explain the meaning of torque and describe its relationship to how levers work.
Use different combinations of weights to balance a lever.
Apply an understanding of rotational equilibrium to determine an unknown mass.

37. Centripetal Force
Any force that causes an object to move in a circle is called a centripetal force.
A centripetal force is always perpendicular to an object’s motion, toward the center of the circle.

38. Centripetal force and direction
Whether a force makes an object accelerate by changing its speed or by changing its direction or both depends on the direction of the force.

39. Centripetal force and direction
Imagine tying a ball to the end of a string an twirling it in a circle over your head.
The string exerts the centripetal force on the ball to move it in a circle.
The direction of the centripetal force changes as the object moves around you.

40. Centripetal force and inertia
If you give a ball on a string an initial velocity to the left at point A, it will try to keep moving straight to the left.
But the centripetal force pulls the ball to the side.

41. Centripetal force and inertia
A short time later, the ball is at point B and its velocity is 90 degrees from what it was.
But now the centripetal force pulls to the right.
The ball’s inertia makes it move straight, but the centripetal force always pulls it towards the center.

42. Centripetal force and inertia
Notice that the velocity is always perpendicular to the string and therefore to the centripetal force.
The centripetal force and velocity are perpendicular for any object moving in a circle.

43. Newton’s 2nd law and circular motion
An object moving in a circle at a constant speed accelerates because its direction changes.
How quickly an object changes direction depends on its speed and the radius of the circle.
Centripetal acceleration increases with speed and decreases as the radius gets larger.

44. Centripetal Acceleration
Acceleration is the rate at which an object’s velocity changes as the result of a force.
Centripetal acceleration is the acceleration of an object whose direction and velocity changes.

45. Newton’s 2nd law and circular motion
Newton’s second law relates force, mass, and acceleration.
The strength of the centripetal force needed to move an object in a circle depends on its mass, speed, and the radius of the circle.

46. Newton’s 2nd law and circular motion
Newton’s second law relates force, mass, and acceleration.
The strength of the centripetal force needed to move an object in a circle depends on its mass, speed, and the radius of the circle.
1. Centripetal force is directly proportional to the mass. A 2-kg object needs twice the force to have the same circular motion as a 1-kg object.

47. Newton’s 2nd law and circular motion
2. Centripetal force is inversely proportional to the radius of its circle.
The smaller the circle’s radius, the greater the force. An object moving in a 1 m circle needs twice the force it does when it moves in a 2 m circle at the same speed.

48. Centrifugal force?
Have you ever noticed that when a car makes a sharp turn, you are pushed toward the outside edge of the car?
This apparent outward force is called centrifugal force.
While it feels like there is a force acting on you, centrifugal force is not a true force.

49. Inertia and circular motion
Suppose a box is in the center of the bed as a truck travels along a straight road.
The box and the truck are both moving in a straight line.
If the truck suddenly turns to the left, the box tries to keep moving in that same straight line.
While it seems like the box is being thrown to the right side of the truck, the truck is actually turning under the box.

50. Universal Gravitation and Orbital Motion
Sir Isaac Newton first deduced that the force responsible for making objects fall on Earth is the same force that keeps the moon in orbit.
This idea is known as the law of universal gravitation.
Gravitational force exists between all objects that have mass.
The strength of the gravitational force depends on the mass of the objects and the distance between them.

51. Gravitational force
The force of gravity between Earth and the Sun keeps Earth in orbit.

52. Gravitational force
You notice the force of gravity between you and Earth because the planet’s mass is huge.
Gravitational forces tend to be important only when one of the objects has an extremely large mass, such as a moon, star, or planet.
If you tell a person on the north pole and one on the south pole to point down, they will be pointing in opposite directions

53. The gravitational force
The force of gravity between two objects is proportional to the mass of each object.
The distance between objects, measured from center to center, is also important when calculating gravitational force.

54. Law of universal gravitation
Newton’s law of universal gravitation gives the relationship between gravitational force, mass, and distance.
The gravitational constant (G) is the same everywhere in the universe (6.67 × 10–11 N·m2/kg2)

55. Law of universal gravitation
The gravitational force of Earth on the Moon has the same strength as the gravitational force of the Moon on Earth.

57. Calculating
Use the following information to calculate the force of gravity between Earth and the Moon: mass of Earth: 5.97 × 1024 m; mass of the Moon: 7.34 × 1022 kg; distance between Earth’s and the Moon’s centers: 3.84 × 108 m.
Looking for: … the force of gravity between Earth and the Moon.
Given: … Earth’s (5.97 × 1024 kg) and the Moon’s masses (7.34 × kg) and the distance between their centers (3.84 × 108 m).
Relationships: Use: F = m1m2 r2
Solution: F = (5.97 × 1024 kg) (7.34 × 1022 kg) = 1.99 X 1020 N
(3.84 × 108 m)2

58. Satellites
A satellite is an object that circles around another object with gravity providing the centripetal force.
Artificial satellites that orbit, or revolve around, Earth include the Hubble Space Telescope.
Earth, its moon, and the other planets are examples of natural satellites.

59. Orbital motion
An orbit can be a circle or an oval shape called an ellipse.
The planets move in nearly circular orbits.
Comets travel in elliptical orbits around the Sun.

60. Center of Mass
There are three different axes about which an object will naturally spin.
The point at which the three axes intersect is called the center of mass.

61. Finding the center of mass
If an object is irregularly shaped, the center of mass can be found by spinning the object and finding the intersection of the three spin axes.
There is not always material at an object’s center of mass.

63. Finding the center of gravity
Closely related to the center of mass is the center of gravity.
The center of gravity of an irregularly shaped object can be found by suspending it from two or more points.
For very tall objects, such as skyscrapers, the acceleration due to gravity may be slightly different at points throughout the object.

64. Center of mass and stability
Objects balance because the torque caused by the force of the object’s weight is equal on each side.

65. Center of mass and stability
For an object to remain upright, its center of gravity must be above its area of support.
The area of support includes the entire region surrounded by the actual supports.
An object will topple over if its center of mass is not above its area of support.

66. Center of mass and stability
An object will topple over if its center of mass is not above its area of support.
If a vector from the center of mass to the center of Earth passes through the area of support, the object will not topple.

68. Sailboat Racing: Vectors to Victory
Racing the 1,022 kilometers from Newport, Rhode Island to Bermuda in a sailboat is one of the great adventures in sailing.
With the Gulf Stream current, giant waves, and unpredictable weather, staying on course is difficult
Before the boat ever starts the race, the planning begins.