Question

Introduction: In this activity a ball is suspended by a piece of string above the floor. The ball swings back and forth as if it were a pendulum. You wll find a mathematical model which represents the position of the ball with respect to time. We are going to ignore resistance, which would have an impact on the maximum height of the ball over time. I. Collecting the data: One student started the ball swinging back and forth as if it were a pendulum. Once the student was able to maintain a periodic motion and a consistent maximum height, the other class members took the following measurements The height of the ball above the ground at its equilibrium position (when the string is vertical) is 0.33 ft The height of the ball above the ground at its highest point is 1.73 ft. .The horizontal distance from the equilibrium point to the highest position of the ball is 2.26 ft The length of time to complete 5 swing cycles (a swing cycle is when the bal returns to the same position moving in the same direction) is 4.80 seconds. In the diagram to the right, the equilibrium position and the maximum height positions of the ball on the right and left are shown on a set of coordinate axes Write the coordinates of ( ) each position.

Answer 1

I. The origin is defined on the ground right below the equilibrium position of the ball.

Therefore, the x coordinate of the equilibrium position is 0. Since it is at a height of 0.33 ft, the y coordinate is 0.33ft. The coordinate is thus (0,0.33)ft

For the maximum height position on the right, it is at a horizontal distance of 2.26ft. Since the coordinate system is defined with right as positive, the x coordinate is +2.26ft. Since it is at a height of 1.73 ft, the y coordinate is 1.73ft. The coordinate is thus (2.26, 1.73)ft

For the maximum height position on the left, it is at a horizontal distance of 2.26ft. Since the coordinate system is defined with right as positive, the x coordinate is -2.26ft. Since it is at a height of 1.73 ft, the y coordinate is 1.73ft. The coordinate is thus (-2.26, 1.73)ft

II. Since five swing cycles take 4.8s, one cycle takes 4.8/5=0.96s

Since the ball is undergoing sinusoidal motion, the time taken to go from maximum height position to equilibrium position is the same as the time taken to go from equilibrium to max height position.

Therefore,

1.At equilibrium- 0s

x(0)=0

2. At farthest point to right- 0.96/4=0.24s

x(0.24s)=2.26ft

3. At equilibrium- 0.24*2=0.48s

x(0.48s)=0

4. At farthest point to the left- 0.24*3=0.72s

x(0.72s)=-2.26ft

5. At equilibrium- 0.96s

x(0.96s)=0

The motion when graphed is sinusoidal.

Since the y coordinate is 0 when the x coordinate is 0, it should be a sin and not a cos.

Assuming we are fitting to $x=Acos(2\pi t/T)+Bsin(2\pi t/T)$ ,

A=0 and B=2.26ft. Period is 0.96s.

Therefore, $x(t)=2.26sin(2\pi t/0.96)ft=2.26sin(6.54t)ft$

IV. If you do the steps outlined in your calculator, you will get the equation for y in terms of x as:

with y<2.85

So, each swing of the ball follows an arc of a circle with radius 2.52 (square root of 6.3504) and centred at (0, 2.85)ft

Your graphing screen will look like:

The above equation can also be obtained analytically by considering the fact that the length of the string remains the same in any position and calculating the x and y coordinate at any random position of the ball in terms of this length using the Pythagoras equation.