We can better understand and visualize many aspects of simple harmonic motion by studying its relationship to uniform circular motion. Figure 13.17 is an overhead view of an experimental arrangement that shows this relationship. A ball is attached to the rim of a turntable of radius A, which is illuminated from the side by a lamp. The ball casts a shadow on a screen. We find that as the turntable rotates with constant angular speed, the shadow of the ball moves back and forth in
simple harmonic motion.
Figure 13.17 An experimental setup for demonstrating the connection between simple harmonic motion and uniform circular motion. As the ball rotates on the turntable with constant angular speed, its shadow on the screen moves back and forth in simple harmonic motion.
Consider a particle located at point P on the circumference of a circle of radius A, as shown in Figure 13.18a, with the line OP making an angle φ with the x axis at t = 0. We call this circle a reference circle for comparing simple harmonic motion and uniform circular motion, and we take the position of P at t = 0 as our reference position. If the particle moves along the circle with constant angular speed ω until OP makes an angle θ with the x axis, as illustrated in Figure 13.18b, then at some time t > 0, the angle between OP and the x axis is θ = ωt + φ. As the particle moves along the circle, the projection of P on the x axis, labeled point Q, moves back and forth along the x axis, between the limits x = ± A.
Figure 13.18 Relationship between the uniform circular motion of a point P and the simple harmonic motion of a point Q. A particle at P moves in a circle of radius A with constant angular speed ω. (a) A reference circle showing the position of P at t = 0. (b) The x coordinates of points P and Q are equal and vary in time as x = A cos(ωt + φ). (c) The x component of the velocity of P equals the velocity of Q. (d) The x component of the acceleration of P equals the acceleration of Q.
Note that points P and Q always have the same x coordinate. From the right triangle OPQ, we see that this x coordinate is
This expression shows that the point Q moves with simple harmonic motion along the x axis. Therefore, we conclude that
simple harmonic motion along a straight line can be represented by the projection of uniform circular motion along a diameter of a reference circle.
We can make a similar argument by noting from Figure 13.18b that the projection of P along the y axis also exhibits simple harmonic motion. Therefore, uniform circular motion can be considered a combination of two simple harmonic motions, one along the x axis and one along the y axis, with the two differing in phase by 90°.
This geometric interpretation shows that the time for one complete revolution of the point P on the reference circle is equal to the period of motion T for simple harmonic motion between x = ± A. That is, the angular speed ω of P is the same as the angular frequency ω of simple harmonic motion along the x axis (this is why we use the same symbol). The phase constant φ for simple harmonic motion corresponds to the initial angle that OP makes with the x axis. The radius A of the reference circle equals the amplitude of the simple harmonic motion.
Because the relationship between linear and angular speed for circular motion v = rω is (see Eq. 10.10), the particle moving on the reference circle of radius A has a velocity of magnitude ωA. From the geometry in Figure 13.18c, we see that the x component of this velocity is -ωA sin(ωt + φ). By definition, the point Q has a velocity given by dx/dt. Differentiating Equation 13.31 with respect to time, we find that the velocity of Q is the same as the x component of the velocity of P. The acceleration of P on the reference circle is directed radially inward toward O and has a magnitude v2/A = ω2A. From the geometry in Figure 13.18d, we see that the x component of this acceleration is -ωA cos2(ωt + φ).This value is also the acceleration of the projected point Q along the x axis, as you can verify by taking the second derivative of Equation 13.31.