Free oscillations of a spring pendulum

What Nour says

1. 1. The spring pendulum and its oscillations

   Today, the whole chapter: "Free oscillations of a spring pendulum", Physics, Year 4, Mathematics section.

   We will build the drawing step by step, and each part ends with a short question for you.

   Let's start with the spring pendulum (pendule élastique).

   A spring pendulum is a spring with one end fixed, and a mass attached to the other end.

   That is, the whole system can stretch or compress and then move back and forth.

   Here is the spring, attached to the wall, lying on a horizontal surface.

   At the free end, we attach a solid mass, called m, that can slide without friction.

   We label its center G, the point we will track as it moves.

   We choose an axis x, horizontal, with origin O at the position of G at rest.

   At rest, the spring has its natural length, and G sits exactly at O.

   Now we pull the mass away from O and let it go, with no push, no initial speed.

   The mass starts moving back and forth around O, all on its own.

   This motion is called oscillations libres non amorties, free undamped oscillations.

   That is, the system oscillates by itself, with no outside force driving it and no energy loss.

   At each instant t, the position of G is given by x of t, measured from O.

   When x is positive, the spring is stretched, and the mass sits to the right of O.

   When x is negative, the spring is compressed, and the mass sits to the left of O.

   The mass keeps moving between two extreme points, one on each side of O.

   These extreme positions have equal distance from O, and that distance is called the amplitude.

   We write this amplitude x_m, and it stays the same throughout the motion, since there is no damping.

   The motion also repeats itself after a fixed time, over and over.

   We call that fixed time the natural period, written T0, in seconds.

   Here is why the motion repeats: the spring always pulls the mass back toward O.

   When the mass passes O, it still has speed, so it overshoots to the other side.

   Then the spring pulls it back again, and the cycle continues forever, in theory.

   Now let's look at a worked example, close to what you may see in the bac.

   A spring pendulum oscillates, and we record its elongation x of t every small time step.

   The data shows the mass returns to the same position and speed every 0.40 seconds.

   This repeating time is exactly the natural period, so T0 equals 0.40 seconds.

   The data also shows the mass reaches a maximum elongation of 0.05 meters from O.

   So the amplitude of this motion is x_m equals 0.05 meters.

   We now frame the two key results we found, T0 and x_m.

   Let's summarize this section with the key idea of this motion.

   The spring pendulum oscillates freely, with a constant amplitude and a constant natural period T0.

   No energy enters or leaves the system, since there is no friction and no driving force.

   This free, undamped motion repeats itself forever, always between the same two extreme positions.

   In this example, is the amplitude x_m equal to 0.05 m or to 0.4 m?

2. 2. Fundamental law of dynamics and the differential equation

   Now let's move to Newton's second law (relation fondamentale de la dynamique) and the differential equation.

   We use Newton's second law (relation fondamentale de la dynamique).

   That is, the sum of all forces on the mass equals its mass times its acceleration.

   Here is the spring pendulum again, mass m, position x measured from O.

   We stretch it to a position x, and let it slide without friction.

   Two forces act on the mass: its weight, and the normal reaction from the surface.

   These two forces are vertical, and they cancel each other out exactly.

   So the only force that changes the motion is the spring's restoring force, called F.

   The spring always pulls or pushes the mass back toward O, opposite to x.

   This force is written F equals minus k times x, where k is the spring stiffness (raideur du ressort).

   That is, k is the stiffness of the spring, how hard it resists stretching, in newtons per meter.

   A stiff spring has a large k, and a soft spring has a small k.

   Now we apply Newton's second law along the x axis.

   The acceleration along x is the second derivative of x of t, written x double dot.

   So the law becomes minus k times x equals m times x double dot.

   We rearrange this into the standard form of the differential equation.

   This is the differential equation of the free undamped spring pendulum.

   Here is why this equation gives oscillations: its solution is a sine function of time.

   The general solution is x of t equals x_m times cosine of omega_0 t plus phi.

   Here omega_0 is called the proper angular frequency, linked to k and m.

   Let's check this works with a worked example, like in a bac exercise.

   A spring pendulum has mass m equals 0.20 kilograms, and spring stiffness k equals 20 newtons per meter.

   First, we write the differential equation using these values of m and k.

   The coefficient of x gives us omega_0 squared, so we compute omega_0 itself.

   So omega_0 equals this value, in radians per second, the proper angular frequency.

   This value tells us how fast the mass oscillates around O.

   We frame this result as the main outcome of this worked example.

   Let's summarize the key result of this section.

   Newton's second law, applied to the spring pendulum, gives one key equation.

   That equation is x double dot plus k over m times x equals zero.

   Its solution is always a cosine function, with angular frequency omega_0 equal to square root of k over m.

   In the differential equation, what does the coefficient k/m represent?

3. 3. Natural period and its factors

   Now let's study the natural period and what it depends on.

   We already met the natural period T0, the time for one full back and forth cycle.

   T0 is linked directly to the proper angular frequency omega_0 we just found.

   Since omega_0 equals square root of k over m, we can rewrite T0 using k and m directly.

   This formula shows T0 depends only on the mass m and the stiffness k.

   It does not depend on the amplitude x_m, nor on how we start the motion.

   Here is why this makes sense: a heavier mass moves slower, so it takes longer per cycle.

   A stiffer spring pulls harder, so it pushes the mass back faster, shortening the period.

   So T0 grows when m grows, and T0 shrinks when k grows.

   Let's work through an example, similar to a bac question on this topic.

   A spring pendulum has mass m equals 0.50 kilograms and stiffness k equals 50 newtons per meter.

   First, we find omega_0 using its formula, square root of k over m.

   Next, we use T0 equals two pi over omega_0 to find the natural period.

   So the natural period of this pendulum is about this value, in seconds.

   Now suppose we double the mass, keeping the same spring stiffness k.

   The new period becomes T0 prime equals two pi times square root of m prime over k.

   So doubling the mass makes the period longer, as we expected from the formula.

   We frame this comparison as the key outcome of this example.

   Let's summarize what controls the natural period of a spring pendulum.

   The natural period T0 equals two pi times square root of mass over stiffness.

   A larger mass increases T0, and a stiffer spring decreases T0.

   The amplitude of the motion never changes T0, only m and k do.

   If we use a stiffer spring with larger k, does the period T0 increase or decrease?

4. 4. Total energy and its conservation

   Now let's study the total energy of the spring pendulum, and its conservation.

   We call this the total mechanical energy of the system (énergie totale).

   That is, the total energy is the sum of kinetic energy and elastic potential energy.

   Here is the mass again, moving between minus x_m and plus x_m on this axis.

   The kinetic energy depends on the mass m and its speed v, the velocity of G.

   The elastic potential energy depends on the stiffness k and the position x.

   Adding them gives the total energy formula of the spring pendulum.

   In free undamped oscillations, this total energy E stays constant over time.

   That is, energy keeps changing form, between kinetic and elastic, but the total never changes.

   Here is why: at the extreme positions, x equals plus or minus x_m, the speed is zero.

   There, all the energy is elastic potential energy, and kinetic energy is zero.

   At O, the position x is zero, so all potential energy is gone.

   There, the speed reaches its maximum value, and all energy is kinetic energy.

   So energy flows back and forth between these two forms, but E stays the same.

   Let's apply this to a worked example, like in a bac exercise.

   A spring pendulum has mass m equals 0.20 kilograms, stiffness k equals 20 newtons per meter, amplitude x_m equals 0.10 meters.

   First, we find the total energy E, using the position x_m where speed is zero.

   So the total energy of this spring pendulum is this value, in joules.

   Next, we find the maximum speed v_max, reached when the mass passes through O.

   Since all the energy becomes kinetic at O, we write E equals one half m v_max squared.

   Computing this square root gives the maximum speed of the mass.

   So the mass reaches a maximum speed of this value, in meters per second, at O.

   Let's summarize the key idea about energy in the spring pendulum.

   The total energy E equals kinetic energy plus elastic potential energy, and it stays constant.

   Energy is maximum kinetic at O, and maximum potential at the extreme positions x_m.

   This conservation of energy is a direct result of having no friction and no driving force.

   At the extreme position x_m, is the speed of the mass zero or maximum?

5. 5. Damped regimes: pseudoperiodic and aperiodic

   Finally, let's study damped regimes: pseudo-periodic and aperiodic regimes (régimes pseudopériodique et apériodique).

   That is, real pendulums lose energy to friction, so the oscillations slowly die out.

   In the pseudoperiodic regime, the mass still swings back and forth, but each swing gets smaller.

   The curve looks like a cosine, but its amplitude shrinks with time, inside a shrinking envelope.

   In the aperiodic regime, friction is so strong that the mass returns to O without oscillating at all.

   It just slides back to rest, slowly, with no back and forth motion.

   Here is why this happens: friction removes mechanical energy from the system, little by little.

   With weak friction, the spring force still wins enough times to cause several swings.

   With strong friction, the motion loses too much energy to swing even once.

   In both cases, the total energy E is no longer constant: it keeps decreasing.

   Let's look at an example close to a bac question on this topic.

   A pendulum is released from x_m equals 0.08 meters, in a fluid that causes friction.

   The recorded curve shows the mass crossing O several times, with shrinking peaks each time.

   Since the mass still oscillates, but with decreasing amplitude, this is the pseudoperiodic regime.

   We frame this identification as the result of the example.

   Now suppose instead the fluid is much thicker, with strong friction on the mass.

   The mass released from the same position just creeps back to O, with no crossing at all.

   This second case is the aperiodic regime, since there is no oscillation at all.

   Let's summarize the key idea about damped regimes.

   Friction makes the total energy decrease over time, unlike the free undamped case.

   The pseudoperiodic regime still shows oscillations, but with shrinking amplitude each cycle.

   The aperiodic regime shows no oscillation at all, just a slow return to rest.

   Does the mass cross O more than once in the aperiodic regime?

   That completes the chapter on free oscillations of a spring pendulum.

   We moved from the basic setup to the equation of motion, the period, the energy, and damping.

   Keep practicing with the two exercises to master this chapter for the bac.