Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second time derivatives of the position, in the special case of one dimensional motion. What is the generalization to motion in two or three dimensions? Simpl that the position must now be described b a vector r(t) and the velocit an acceleration are derivatives with respect to time of this vector: Velocit and Acceleration Velocit: v = dr dt Acceleration: a = dv dt = d2 r dt 2 Here the derivative has its usual definition. For eample: dr dt = lim Δt 0 r(t + Δt) r(t). Δt As alwas, a vector equation is three equations, one for each component. Thus v = dr dt means v = d/dt v = d/dt v z = dz/dt Constant Velocit or Constant Acceleration In one dimension the formula for position of a particle moving at constant velocit was found earlier to be = 0 + vt, where 0 is the position at t = 0. PHY 53 1 Kinematics 2
Generalizing this to two or three dimensions is eas: replace the position and velocit b the corresponding vectors: Constant velocit (general) r = r 0 + vt This vector equation is, as alwas, three component equations: = 0 + v t = 0 + v t z = z 0 + v z t The two vectors r 0 and v lie in a particular plane. If we choose that to be the - plane, then the z components of all the vectors in the vector equation will be zero, and we can ignore the third component equation altogether. This is an eample of how choosing the coordinate sstem can often simplif the calculations. Appling the same procedure to the formulas for constant acceleration, we obtain: Constant Acceleration (General) r(t) = r 0 + v 0 t + 1 2 at2 v(t) = v 0 + at The third formula we found in one dimension, relating speed directl to position, involves the scalar product of two vectors, to be discussed later. Gravit Near Earth's Surface As everone knows, an object dropped from rest at a point slightl above the earth s surface falls down; i.e., it eperiences an acceleration directed toward the surface. Galileo discovered b eperiment two remarkable facts about this: 1. If the effects of the air and other possible influences are neglected, the acceleration is constant. 2. The acceleration is the same for all objects. That the acceleration is constant is an approimation, requiring that the distance the object is above the earth s surface be small compared to the earth s radius. The eact situation will be discussed later when we stud gravit more closel. But that the acceleration is the same for all objects is a fundamental propert of gravit. PHY 53 2 Kinematics 2
We call this the acceleration of gravit and give it the special smbol g. It is directed downward (toward the center of the earth). Its magnitude near earth s surface varies slightl from place to place on the earth, but a useful value is 9.80 m/s 2. In making rough calculations, and on eams in this course, one can use g = 10 m/s 2. In our later discussion of gravit the quantit g will be called the gravitational field. The motion of an object subject onl to gravit near earth s surface is an important case of constant acceleration. To get the position and velocit we merel set a = g : r = r 0 + v 0 t + 1 2 gt2 v = v 0 + gt To use these equations it helps to set up a coordinate sstem and use the components of the vectors. In the sstem shown we have g g = g and the other components of g are zero. If we also choose the orientation of the aes so that v 0 has no z-component, then the motion will take place in the - plane (or a plane z Surface parallel to it, if r 0 has a non-zero z-component). All the z-components are either zero or constant, so we have in effect a 2-D problem. With this coordinate sstem, the equations for the and components of position are: (t) = 0 + v 0 t Gravit near earth s surface (t) = 0 + v 0 t 1 2 gt2 v (t) = v 0 v (t) = v 0 gt These four equations describe the motion of an object, subject onl to gravit, moving near the earth's surface.. Motion of Projectiles An important case of this motion is that of objects thrown or shot into the air, i.e., projectiles. We neglect effects of air resistance, so onl gravit influences the motion. The relevant equations are the ones given above. It is common to give the initial velocit in terms of the speed and the elevation angle above the earth s surface. This means the initial velocit components must be epressed in terms of these quantities. PHY 53 3 Kinematics 2
The initial situation is as shown. The initial velocit components are v 0 v 0 = v 0 cosθ, v 0 = v 0 sinθ. The equations describing the motion then become: θ (t) = 0 + v 0 cosθ t Projectile motion near earth s surface (t) = 0 + v 0 sinθ t 1 2 gt2 v (t) = v 0 cosθ v (t) = v 0 sinθ gt The equation for the trajector can be obtained b eliminating t between the equations for and. If the projectile starts from the origin (i.e., 0 = 0 = 0 ) the result is = tanθ g 2 2(v 0 cosθ) 2. This equation describes a parabola in the - plane. An eample is shown. ma v 0 Range The total distance traveled horizontall before returning to the initial height is the range, denoted b R. It is found b setting = 0 in the trajector equation above and solving for the non-zero value of. The result is R = v 2 0 sin 2θ. g One sees from this formula that for given v 0 the range is largest if θ = 45, and that the range is the same for an two angles whose sum is 90. These facts were discovered b Galileo. PHY 53 4 Kinematics 2
The maimum height reached occurs when the vertical component of the velocit is (momentaril) zero, so we find b setting d/d = 0 and solving for. One finds ma = (v 0 sinθ)2 2g. Motion in a Circle Motion for which the acceleration is zero, or for which it is alwas along the same line as the velocit, is motion in a straight line. It follows that: If a particle moves in a curved path it must have a component of acceleration perpendicular to the velocit. This perpendicular component of acceleration does not change the speed, but it does change the direction of the velocit. There might also be a component of acceleration along the line of the velocit which changes the speed. If the path followed is a circle, these acceleration components are relativel simple. The change of direction is brought about b the radial acceleration, directed radiall toward the center of the circle. If the speed changes, there is also an acceleration tangent to the circle, called the tangential acceleration. We will analze circular motion in detail. Cases in which a particle s trajector is a circle occur frequentl, especiall in the description of objects rotating about some ais. Let the particle move in a circle of radius R about the origin in the - plane, as shown. The equation of the circle is 2 + 2 = R 2. Since R is constant, the particle s position at an time can specified b giving the value of the angle θ at that time. The position vector of the particle is given b r(t) = i Rcosθ(t) + j Rsinθ(t). This can convenientl be written in terms of a unit vector ˆr parallel to r (i.e., radiall outward from the origin to the particle). We write r = Rˆr, where ˆr = icosθ + jsinθ. As the particle moves the angle θ changes, so ˆr is a function of time: its magnitude is constant, but its direction changes.. To get the velocit we take the time derivative of r. Since R is constant: v = dr dt = R dˆr dt = R i d dt cosθ + j d dt sinθ = R dθ [ isinθ + jcosθ]. dt R θ PHY 53 5 Kinematics 2
The quantit in [ ] in the last term is another unit vector, with direction tangent to the circle in the counter-clockwise sense (the direction of increasing θ ). We give it a smbol: ˆθ = isinθ + jcosθ. The time derivative dθ /dt is called the angular velocit, denoted b ω: Angular velocit ω = dθ /dt We have derived an important formula for the velocit vector: Velocit in circular motion v = Rω ˆθ If the angular velocit ω is positive (θ is increasing) the velocit v is tangent to the circle going counter-clockwise. If ω is negative, the velocit is tangent clockwise. Later we will see that it is useful to treat the angular velocit as a vector perpendicular to the plane of the motion. Here the sign of ω is all we need to specif the sense of the rotation. The linear speed is related to the angular speed b a ver useful relation: Speed in circular motion v = R ω To get the particle s acceleration we take the time derivative of v. The unit vector ˆθ also varies with time. If the angular velocit ω varies with time (making the speed change) the acceleration will consist of the two terms referred to above the radial and tangential accelerations. We define the angular acceleration as the time derivative of the angular velocit: Angular acceleration α = dω /dt Then working out a = dv/dt we find PHY 53 6 Kinematics 2
Radial and tangential acceleration a = a r + a t where a r = ω 2 R ˆr a t = αr ˆθ We will look in more detail at these two tpes of acceleration and their effects. Radial Acceleration. ( a r = ω 2 Rˆr ) The negative sign in the formula shows that a r is opposite to ˆr, so it is alwas toward the center of the circle. Its effect is onl to change the direction of the velocit as the particle moves around the circle; it does not change the speed. Its magnitude can be written in two equivalent forms: Radial acceleration magnitude a r = ω 2 R = v 2 /R The radial acceleration is alwas toward the center, so it is often called the centripetal (center-seeking) acceleration. This archaic and unnecessar term will not be used here. Tangential Acceleration. ( a t = αrˆθ ) This acceleration changes onl the speed. It is parallel to the velocit if the speed is increasing. Its magnitude can be written in two forms: Tangential acceleration magnitude a t = R α = dv/dt The derivative in the last term is the rate of change of the speed, not of the velocit. If the motion is one at constant speed, it is called uniform circular motion. In that case the speed and the angular velocit are constant, so the tangential acceleration is zero. But there is alwas a radial acceleration which changes the direction of the velocit. Because motion in a circle requires an acceleration, it does not happen without some eternal influence to cause that acceleration, so it is not a natural uncaused motion, which people thought it was until the 17 th centur. The were misled b the apparent motion of the celestial bodies going around the earth once a da, an illusion caused b the rotation of the earth. PHY 53 7 Kinematics 2