A basic classical example of simple harmonic motion is the simple pendulum, consisting of a small bob and a massless string. In a vacuum with zero air resistance, such a pendulum will continue to oscillate indefinitely with a constant amplitude. However, the amplitude of a simple pendulum oscillating in air continuously decreases as its mechanical energy is gradually lost due to air resistance. To this end, it is generally perceived that the main role in the dissipation of mechanical energy is played by the bob of the pendulum, and that the string’s contribution is negligible. The purpose of this research is to experimentally investigate the merit of this assumption. Thus, we experimentally investigate the damping of a simple pendulum as a function of its string diameter and compare that to the contribution from its bob. We find out that although in some cases the effect of the string might be small or even negligible, in general the string can play a significant role, and in some cases even a greater role on the damping of the pendulum than its bob.
Perhaps the simplest oscillating system is a small object attached to a string of negligible mass, known as simple pendulum. If the amplitude of oscillations is small, the pendulum oscillates with a period T which is independent of the amplitude and is given by
T = 2 π L g (1)
where L is the length of the pendulum and g is the acceleration due to gravity (9.80 m/s2). In the absence of frictional losses, the hypothetical pendulum would oscillate indefinitely. However, the amplitude of oscillations of any real undriven mechanical pendulum, including simple pendulum, continuously decreases as a result of frictional losses, mainly due to air resistance while its period and frequency remain constant. To this end, the general assumption is that the drag force due to the air resistance on the bob of the pendulum is the cause of its damping, and normally the air resistance on the string of the pendulum is assumed to be negligibly small.
In order to be able to measure the gravitational acceleration very accurately, Nelson and Olsson [
Motivated by the work of Nelson and Olsson and by Dunn, we decided to further investigate the effect of string on damping of a simple pendulum. To do so, we experimentally studied the contribution to the damping of a simple pendulum from strings of various diameters.
When a solid object moves in a fluid, the magnitude of the drag force is in general a function of the speed of the object, F = F ( v ) . Expanding this function in a Taylor series about v = 0 , we have
F = F ( 0 ) + F ′ ( 0 ) 1 ! v + F ′ ′ ( 0 ) 2 ! v 2 + ⋯ (2)
However, the constant term is zero because when v = 0 there is no drag force. Therefore,
F = c 1 v + c 2 v 2 + ⋯ (3)
where c 1 , c 2 , ⋯ are constants. For small speeds, we can approximate the drag force by the first-order term and neglect the second- and higher-order terms. Experimentally it is found that for a relatively small object moving in air with speeds less than about 24 m/s, the force of air resistance is proportional to the first power of speed. For higher speeds, but below the speed of sound, the force is proportional to the square of the speed [
The common practice, however, is to write the magnitude of the drag force on an object moving in a fluid as
F = 1 2 C D ρ A v 2 (4)
where ρ is the density of the fluid , v is the velocity of the object, and A is the frontal cross-sectional area of the object, i.e., the cross-sectional area of the object perpendicular to its direction of motion. The unitless parameter C D is the drag coefficient. Drag coefficient depends on the shape of the object and the Reynolds number,
R e = ρ L | v | μ (5)
where L is a characteristic diameter or linear dimension of the object, such as diameter of a sphere, and μ is the absolute or dynamic viscosity of the fluid. For R e of the order of about 1200 or less, the drag coefficient C D is asymptotically proportional to R e − 1 , which means that the drag force is a proportional to the first power of velocity [
Therefore, for a simple pendulum moving with small speeds (long pendulum), the force of air resistance on its bob, F b , is proportional to its velocity,
F b = − c v (6)
where c is a constant, independent of velocity, but depends on the shape and frontal cross-sectional area of the bob. We now calculate the force of air resistance on the string of the pendulum.
Consider an element of the string of a pendulum of length d r located at a distance r from the support point and moving with velocity v , as shown in
d F s = k ( D d r ) v (7)
where k is a constant. But since v = r θ ˙ , where θ ˙ is time derivative of θ , we obtain
d F s = k D θ ˙ r d r (8)
Since this drag force is perpendicular to the string, its torque about the support point is
d τ s = r d F s = k D θ ˙ r 2 d r (9)
Integration of this equation over the length of the string, L, gives the total torque on the string,
τ s = k D θ ˙ ∫ 0 L r 2 d r = k L 3 D 3 θ ˙ (10)
Derivation of the equation of motion of the simple pendulum with a linear drag force is trivial, however, we present it here for completeness of the discussion.
The equation of motion of the spring is
∑ τ = I α (11)
where all torques are calculated relative to the support point, α is the angular acceleration of the pendulum about this point, and I = m L 2 is the moment of inertia of the pendulum about the support point (string has negligible mass). Using
m L 2 θ ¨ = − m g L sin θ − τ s − F b L (12)
where τ s is given by Equation (10). The third term on the right hand side of this equation is the torque caused by the force of air resistance on the bob of the pendulum, in which F b is given by Equation (6), i.e.,
F b = c v = c L θ ˙ (13)
Therefore, after some simplifications, Equation (12) becomes
θ ¨ + κ θ ˙ + g L sin θ = 0 (14)
where the damping constant κ is defined by
κ = k L D 3 m + c m (15)
which has the units of s−1 in the SI system. The first term on the right hand side of this equation is the contribution to the damping of the pendulum due to its string and the second term is that due to its bob. Finally, if the amplitude of oscillations is small, we have s i n θ ≈ θ , and Equation (14) reduces to
θ ¨ + κ θ ˙ + g L θ = 0 (16)
Equation (16) is a homogeneous second order linear differential equation with constant coefficients, whose solution is straightforward. We fist construct the auxiliary equation,
q 2 + κ q + g L = 0 (17)
which has the solutions
q = − κ ± κ 2 − 4 g L 2 (18)
Since air resistance is small, we have
κ 2 < 4 g L (19)
and Equation (18) becomes
q = − κ ± i 4 g L − κ 2 2 (20)
Then, the general solution of the differential equation of motion (16) is
θ = e − κ t / 2 [ A cos ( ω t ) + B sin ( ω t ) ] (21)
where A and B are constants to be determined by the initial conditions, and the angular frequency ω is given by
ω = g L − κ 2 4 (22)
Applying the initial condition θ ( t = 0 ) = θ 0 and θ ˙ ( t = 0 ) = 0 , we obtain
θ = θ 0 e − κ t / 2 [ cos ( ω t ) + κ 2 ω sin ( ω t ) ] (23)
Therefore, the amplitude of the oscillations decreases exponentially with time according to
θ = θ 0 e − κ t / 2 (24)
Throughout our experiment we used a steel ball of mass 485 g and diameter 50.9 mm as the bob of our simple pendulum. For string we used monofilament nylon fishing lines of various diameters. During the entire experiment, the total length of the pendulum was 261.4 cm, 258.9 of which was the length of the string. Although the thickness of each string was consistent throughout its length, we measure the diameter 8 times along its length while the string was under load.
With each string, we started with an amplitude of 30 cm for the oscillations of the bob and measured the time interval for every 1 cm decrease in the amplitude
Test Line | 8-lb | 12-lb | 20-lb | 30-lb | 50-lb |
---|---|---|---|---|---|
Diameter (mm) | 0.250 | 0.279 | 0.428 | 0.570 | 0.718 |
until the amplitude dropped to 10 cm. We note here that since the length of the string is about 259 cm, the linear amplitude of 30 cm corresponds to an angular amplitude of about 6.65 ∘ . Then the error in using the approximation sin θ ≈ θ in Equation (14) is only about 0.2%. We also note that the mass of the heaviest string used in our experiments (the 50-lb test line) was only 1.5 g, which is quite negligible compared to the mass of the bob.
Equation (24) may be written as
ln ( θ θ 0 ) = − κ 2 t (25)
Therefore, a graph of l n ( θ / θ 0 ) versus t should be a straight line with a slope of − κ / 2 .
Because we used the same bob and the same string length in all experiments, according to Equation (15) the graph of the damping constant κ as a function of string diameter D should be a straight line. This is shown in
κ = ( 0.777 ± 0.067 ) D + ( 7.24 ± 0.32 ) × 10 − 4 (26)
in which D is in meters and κ is in s−1. This line is also plotted in the figure.
The first term in Equation (26) is the contribution of the string of the pendulum to its damping ( κ S ), and the second term is that of its bob ( κ B ). Using the diameters of our strings and Equation (26), we can calculate these contributions in our experiments. The results are shown in
In all cases studied, the period of the pendulum was about 3.27 s. Since the initial amplitude of the motion of the bob was 30 cm, this gives an average speed of
String Diameter D (mm) | 0.250 | 0.279 | 0.428 | 0.570 | 0.718 |
---|---|---|---|---|---|
Damping Constant κ (10−4 s−1) | 9.02 | 9.37 | 10.75 | 11.95 | 12.56 |
String Diameter (mm) | ||
---|---|---|
0.250 | 9.18 | 7.24 |
0.279 | 9.41 | 7.24 |
0.428 | 10.57 | 7.24 |
0.570 | 11.67 | 7.24 |
0.718 | 12.82 | 7.24 |
0.367 m/s. With a room-temperature density of 1.204 kg/m3 [
The results of
In conclusion, the results of this investigation show that the string of a simple pendulum plays a significant role, and in some cases a more important role, in damping the pendulum than its bob. To the best of our knowledge, this effect has not been taken into account in the discussions of damping of pendulums in the literature.
This work was supported by a URAP grant from the University of Wisconsin- Parkside.
Mohazzabi, P. and Shankar, S.P. (2017) Damping of a Simple Pendulum Due to Drag on Its String. Journal of Applied Mathematics and Physics, 5, 122-130. http://dx.doi.org/10.4236/jamp.2017.51013