CHAPTER 3

MOTION AND FORCES IN MORE THAN ONE DIMENSION

By :Ari Dewayani (12316244010)Navis Aininah (12316244016)

One thing wa clear about beginning and end points

the path of a projectiles wa difficult for the acients

to trace

Solved :

Fig 3-1

connon

Fig. 3-1. A cannon or a caapult must direct a procectile at an angle above the horizontal line to hit a distant object on the same level.

Gallileo’s

Projectile falls verticaly as moves horizontals

In terms of forces produce curvacture aided the discription of curved motion in general

3-1 Direct Quantities : Vectors

The east-west line The North-South line The vertical or artitude line

Direction for a fairly small region of the earth’s surface

Directed Quantity displacement tof the point of a pencil from one place on a notebook page to another is called vector.

Displacement : relative position depends only on initial and final positions and is independent of the path beteween.Fig 3-2 Two actual displacements are made, but they result in the single vector OP. Vector OP is the sum of vectors OA dan OP.

1.4 in

1 in

1 in A

E

P

O

Displacement from O to P

Fig 3-4

A line segment whose length represents its magnitude in accord with some predetermined scale, and whose direction is that of the force. The two may be added as though they were displacements and the magnitude of the vector sum or resultant.On the first law of motion, since equilibrium is defined as an absence of accelertion.

2 lb

C

B

20 lb

A

D

Two forces acting at the same point but in different direction represented in the figure.

A detailed diagram showing just how forces are distributed in a given equilibrium in nonetheless extremely useful in practice. (Fig 3-5)

The physical problem supporting a 100lb object by two strings in the manner shown in (a) maybe analyzed by a force diagram (b) The resultant sum R of the forces T1and T2 exerted by the strings must be equal and opposite to weight w.

100lb

30° 60°

R

T1

T2

W

Figure 3-6 which is velocity diagram, is applicable only to individual instans of time, no indicationshow long those conditions prevail.

In practice, a single velocity can be specified by giving its magnitude (speed) and appropriate angles for describing its direction, but diagram is invalube for determining the net result of two simultaneous velocities.

3-1 Motion of Projectiles

The correct of Galilleo disregarded fallacious views at once in terms of Gallileo’s discoveries about falling bodies and the most elementary ideas contained in Newton’s Law.Law of inertia a state either of rest or of uniform motion in horizontal direction is then maintaned unaltered.

Gallileo

Work out the parabolic path for a projectile

The horizontal motion was uniform while the

simultaneous downward motion was uniformly accelerated

The important point is that the motion of the horizontally fired ball, neglecting air resistance, is made up teo independent but simultaneous parts, manifested in two perpendicular direction.Actual path o16ft

64ft

144ft

1. Actual velocities different instans2. Other features of the motion

o 40ft 80ft 120ft 160ft

1sec

2sec

3sec

4sec

Position shown are those at the end of the 1st, 2nd, 3rd and 4th second of fall from a tower 256 ft high . Thr path on the left is that of a body which is dropped and the curved (parabolic path of a body fired horizontally at a speed of 40 ft/sec. (Figure 3-7)

Uniform horizontal velocity the ball in its flight, if the cart also moves at steady speed it will catch the ball when the latter returns to the horizontal level from which it was fired. Figure 3-8

Fig. 3-8. Ball projected upward from a cart in uniform motion keeps abreast of the cart as it rises and falls

3-3 Uniform Circular Motion

Huygens’ word “Maintain forever a motion once impressed upon it.”In the absence of forces, however since the earth is curved, the object would progressively from the earth’s surface. (Figure 3-9).

Object fired horizontally from top of tower at various speeds strikes earth at various distance. With unifrom velocity it would recede from the earth surface.

Uniform velocity

For projectiles subject to the inescapable influence of gravity, fired at great horizontal speeds, the effect of the earth curvature would be to make the time of fall a little greather than expected from Galileo’s Law of free fall.

Principle

It should be possible to fire at a bullet with a speed so great that it falls toward the earth at the same rate that the earth’s surface recedes from it bcause of the earth urvature

If there were no air friction and no obstacles, it could continue to go

around earth at a constant distance from its center

Figure 3-10

The distance such a projectile would travel in a time t if there no gravity. Let y represent the distance fallen in this same time, such that the projectile maintains its original distance from the center of the earth.

d is perpendicular to the radius of the earth at the starting point we use theorem of pythagoras, then we get :

d2 = y2 + 2ry

According to Galileo’s of free fall :

y =

Represent the speed by v := α r, so

Summary• The world is three-dimensional, and changes in

motion may include change in direction as well as in speed.

• Directed quantities called vectors• A body in uniform circular motiojn is accelerated

toward the center of the circle although its speed remains constant.

• Cetripetal is required to produce this acceleration

• Galileo was able to achieve the first satisfactory account of projectile motion.

• The rotation of the earth about its axis involves such forces.