Kinematics of a solid in translation

What Nour says

1. 1. Translation motion and position vector

   Today, the whole chapter: "Kinematics of a solid in translation", Physics, Year 3, Mathematics section.

   We will build this step by step, from the basic vectors to full motion equations, with worked examples like the bac.

   Let's start with translation motion, the simplest kind of motion for a solid.

   A solid is in translation when every segment inside it keeps the same direction during the motion.

   This means all points of the solid move the same way, with the same velocity at every instant.

   Look at these two segments of the solid, at two different moments.

   They stay parallel to each other, which is the key sign of translation motion.

   To describe where a point is, we use its position vector.

   The position vector, written OM, that is the arrow from a fixed origin O to the moving point M, locates the point at any time.

   Here is the origin O, fixed in our reference frame, the frame we measure motion from.

   And here is the point M, the moving point we are tracking on the solid.

   The vector OM has coordinates, given by the position of M along the x and y axes.

   As time passes, the point M moves, and the position vector changes length and direction.

   Here is why this vector matters: it is the tool that lets us write down the whole motion.

   If we know x of t and y of t at every instant, we know the full trajectory of the point.

   For a solid in pure translation, every point follows a trajectory of the exact same shape.

   Now let's work through a typical bac-style example on this idea.

   A small cart moves in translation on a straight horizontal rail.

   At time t equals zero second, its position vector from origin O is 2 meters along the rail.

   At time t equals 3 seconds, its position is 11 meters from O, along the same rail.

   We want to find the average velocity of the cart between these two instants.

   The average velocity is the change in position divided by the change in time.

   Here, the change in position is 11 meters minus 2 meters.

   The change in time is 3 seconds minus 0 second.

   So the average velocity equals 9 meters divided by 3 seconds.

   This average velocity tells us how fast the position vector grew, on average, over that time.

   Notice the result has a unit, meters per second, because velocity always needs a unit.

   This vector method works the same way in two dimensions, with x and y together.

   The direction of this average velocity vector is the same as the direction of the displacement.

   So the position vector is our starting tool for describing any translation motion.

   Let's summarize the key idea of this section.

   In translation motion, all points of the solid move with the same velocity at each instant.

   The position vector OM, from a fixed origin to the moving point, describes where that point is at any time.

   Studying one single point of the solid is enough to describe the motion of the whole solid.

   In translation motion, do all points of the solid have the same velocity at a given instant?

2. 2. Curvilinear abscissa and velocity

   Now let's look at a new way to locate a point: the curvilinear abscissa.

   The curvilinear abscissa, written s of t, is the distance measured along the trajectory from a chosen origin point.

   In other words, it is like a curved ruler laid along the path the point follows.

   We pick a reference point on the trajectory, call it O, where the curvilinear abscissa is zero.

   As the point moves forward, its curvilinear abscissa s of t grows, measured along the curve.

   Here the point M has moved along the path, and s of t is the length O M measured on the curve.

   This works for a straight line, a circle, or any curved path a solid in translation can follow.

   Now, from this curvilinear abscissa, we define the linear velocity of the point.

   The velocity v of t is the derivative of s of t with respect to time.

   This velocity tells us how fast the distance traveled along the path is increasing at each instant.

   Here is why we need the derivative: velocity can change constantly, so we look at it instant by instant.

   If s of t grows at a constant rate, the velocity stays the same number at every instant.

   If s of t grows faster and faster, the velocity itself keeps increasing over time.

   Now let's apply this to a bac-style worked example.

   A solid moves along a straight rail, and its curvilinear abscissa is given by a formula.

   Here s is in meters and t is in seconds, following the usual convention.

   We want the velocity of the solid at time t equals 2 seconds.

   First we find v of t by taking the derivative of s of t with respect to time.

   The derivative of 3 t squared is 6 t, and the derivative of 2 t is 2.

   Now we plug in t equals 2 seconds into this velocity formula.

   This gives us the instantaneous velocity, the exact speed of the solid at that single moment.

   Notice this velocity is a number in meters per second, since s is in meters and t in seconds.

   This result differs from average velocity, since here we use one exact instant, not two separated moments.

   In fact, average velocity over an interval only approaches this instant value as the interval shrinks to zero.

   This connects back to the position vector we saw in the first section of this chapter.

   The length of the position vector traveled matches the curvilinear abscissa, when motion is along one path.

   So both tools, the position vector and the curvilinear abscissa, describe the same real motion.

   Let's summarize what we just covered in this section.

   The curvilinear abscissa s of t measures distance traveled along the trajectory from a fixed reference point.

   The velocity v of t is the derivative of s of t, telling us the instant rate of travel.

   This velocity can be constant, increasing, or decreasing, depending on how the solid moves.

   What formula gives the instantaneous velocity from the curvilinear abscissa s of t?

3. 3. Acceleration: tangential and normal components

   Let's move to acceleration, which tells us how velocity changes over time.

   When a solid follows a curved path, its acceleration has two separate parts.

   Here is a curved trajectory, with a point M moving along it.

   The first part is tangential acceleration, that is the part along the direction of motion, tangent to the path.

   Tangential acceleration changes the speed of the point, making it go faster or slower.

   The second part is normal acceleration, that is the part perpendicular to the path, pointing toward the inside of the curve.

   Normal acceleration changes the direction of motion, bending the path of the point.

   The total acceleration vector is the sum of these two perpendicular parts, tangential and normal.

   Here are the formulas for each component, written with the velocity v of t.

   Tangential acceleration is the derivative of speed v with respect to time.

   Normal acceleration equals speed squared divided by R, the radius of curvature of the path.

   Here is why this split matters: a car can speed up and turn at the same time.

   On a straight line, the radius of curvature becomes infinite, so normal acceleration drops to zero.

   That means on a straight path, total acceleration is only the tangential part.

   Now let's solve a bac-style example with both components.

   A solid moves on a circular path of radius 2 meters.

   At a given instant, its speed is 4 meters per second, and it is increasing at 3 meters per second squared.

   We want the normal acceleration and the total acceleration magnitude.

   First, normal acceleration equals speed squared divided by the radius.

   This gives us 8 meters per second squared, pointing toward the center of the circle.

   Next, since tangential and normal acceleration are perpendicular, we use the Pythagorean rule to combine them.

   We plug in 3 meters per second squared and 8 meters per second squared.

   This is the full magnitude of acceleration, combining the speed change and the direction change together.

   Notice both results carry the unit meters per second squared, the standard unit for acceleration.

   This example shows a solid can both speed up and curve, with one acceleration vector describing both effects.

   Let's summarize what we just learned about acceleration.

   Tangential acceleration changes speed, and normal acceleration changes direction, along a curved path.

   The total acceleration combines both, using the derivative of v and the radius of curvature R.

   On a straight line, normal acceleration disappears, and only tangential acceleration remains.

   On a straight line, is the normal acceleration zero or not zero?

4. 4. Uniformly varied rectilinear motion

   Now let's study a special kind of motion called uniformly varied rectilinear motion.

   Uniformly varied rectilinear motion, that is motion on a straight line where acceleration stays constant over time.

   Since the path is straight, the normal acceleration is zero, just like we saw before.

   Here is the velocity plotted against time: it is a straight line, since acceleration is constant.

   The slope of this line is exactly the tangential acceleration, a constant number.

   From constant acceleration, we get the velocity law by integrating acceleration over time.

   Here v_0 is the velocity at time zero, and a is the constant acceleration.

   Integrating once more gives us the position law, telling us where the solid is at each instant.

   Here x_0 is the starting position, at time t equals zero second.

   Here is why this matters: these two laws describe cars, falling objects, and braking trains, all at once.

   When acceleration a is positive, the motion speeds up, and we call it accelerated motion.

   When acceleration a is negative, the motion slows down, and we call it decelerated motion.

   Now let's solve a full bac-style example with these laws.

   A car starts with velocity 10 meters per second and accelerates at 2 meters per second squared.

   Its initial position is zero meter, taken as our reference point.

   We want its velocity and its position at time t equals 4 seconds.

   First, velocity equals v_0 plus a times t.

   We plug in 10 meters per second plus 2 meters per second squared times 4 seconds.

   This gives 18 meters per second as the velocity at that instant.

   Next, position equals x_0 plus v_0 times t, plus one half a times t squared.

   We plug in the numbers and get 56 meters as the position after 4 seconds.

   Both results carry the right units, meters per second for velocity, meters for position.

   This method works the same way for any object moving in a straight line with constant acceleration.

   A falling object near Earth's surface, ignoring air, also follows these same two equations.

   Let's summarize the key points of uniformly varied rectilinear motion.

   In this motion, acceleration stays constant, and the normal part is always zero, since the path is straight.

   Velocity grows or shrinks linearly in time, following v_0 plus a times t.

   Position follows a quadratic law in time, built from x_0, v_0, and the constant acceleration a.

   In this motion, does velocity change linearly or quadratically with time?

5. 5. Sinusoidal rectilinear motion

   Now let's study another special motion, called sinusoidal rectilinear motion.

   Sinusoidal rectilinear motion, that is motion on a straight line where position oscillates back and forth, following a sine function.

   Think of a mass on a spring, moving left and right around a center point, over and over.

   Here is the position x of t, drawn against time: a smooth repeating wave.

   The general formula for position is x_m times the sine of omega times t plus phi.

   Here x_m is the amplitude, the maximum distance from the center point.

   Omega is the angular frequency, and phi is the phase, fixing the position at time zero.

   Velocity and acceleration come from taking derivatives of this position formula, one after another.

   Taking the derivative of position gives velocity, with cosine replacing sine, and an extra factor omega.

   Taking the derivative again gives acceleration, which is just minus omega squared times position itself.

   Here is why this relation matters: it shows acceleration always pulls the point back toward the center.

   When position is at its maximum, acceleration is most negative, pulling the point strongly back.

   When position passes through the center, acceleration becomes zero, for an instant.

   Now let's work through a bac-style example using these formulas.

   A small object oscillates with amplitude 0.2 meter and angular frequency pi radians per second.

   Its phase at time zero is zero radian, so the motion starts at the center.

   We want the velocity of the object at time t equals 0.5 second.

   We start from the velocity formula, x_m times omega times cosine of omega t plus phi.

   We plug in the amplitude, the angular frequency, and the time 0.5 second.

   This result comes directly from the cosine value at that particular instant.

   Notice the unit stays meters per second, since velocity always needs this unit.

   This oscillating motion repeats itself after a fixed time, called the period of the motion.

   The period relates to omega: a larger angular frequency means a shorter period of oscillation.

   Let's summarize the key points of sinusoidal rectilinear motion.

   Position follows a sine function in time, with amplitude x_m, angular frequency omega, and phase phi.

   Acceleration equals minus omega squared times position, always pulling the point back toward the center.

   This motion repeats itself forever, swinging between plus x_m and minus x_m.

   Does acceleration in this motion point toward the center or away from it?

6. 6. Putting it all together: reading a motion graph

   Let's finish by learning how to read a motion graph and recognize what it tells us.

   A motion graph plots a quantity, like position or velocity, against time, on two axes.

   Here is a velocity versus time graph for a solid in translation.

   From zero to 2 seconds, the line rises, so the solid speeds up with constant acceleration.

   From 2 to 3.5 seconds, the line stays flat, so velocity is constant and acceleration is zero.

   A flat velocity line always means uniform motion, with no change in speed.

   Here is why the slope matters: the slope of a velocity graph always gives the acceleration.

   A steep positive slope means strong acceleration, and a flat slope means zero acceleration.

   Now let's apply this reading skill to a bac-style question.

   On this graph, the velocity rises from 0 meter per second to 6 meters per second between 0 and 2 seconds.

   We want the acceleration of the solid during this first phase of motion.

   Acceleration is the slope, so it equals the change in velocity divided by the change in time.

   We plug in 6 meters per second minus 0 meter per second, divided by 2 seconds minus 0 second.

   This gives us an acceleration of 3 meters per second squared during the first phase.

   After 2 seconds, since the slope is flat, the acceleration during that phase equals zero meter per second squared.

   This single graph lets us describe the whole history of the solid's motion, phase by phase.

   The same reading method works for position graphs, where the slope gives velocity instead of acceleration.

   Let's summarize everything we covered in this final section.

   A velocity graph's slope always gives acceleration, and a flat line always means uniform motion.

   Reading these graphs connects every idea from this chapter: position, velocity, and acceleration together.

   With this skill, you can analyze any real motion graph given in a bac exam.

   On a velocity-time graph, what does a flat horizontal line mean for acceleration?

   That completes our chapter on the kinematics of a solid in translation.

   We moved from the position vector, to curvilinear abscissa, to acceleration, and the two special motions.

   Practice the two exercises above to prepare well for your bac exam.