In years past, steam locomotive wheels were driven by a connecting rod attached to a cylinder. Open the Geometer's Sketchpad file below. You will see a simulation representing a toy locomotive wheel. By clicking and dragging the point on the wheel you can see that the motions of the wheel and cylinder can be modeled using trigonometric functions.
Your task is to create an equation in the form y = a sin k(θ - c) + d modeling the motion of the cylinder with respect to the rotational angle of a point on the wheel. To do this, rotate the point, and double click on the data table on the left to collect data.
a) Use this information to create an equation.
b) If you are told that the speed of rotation is 150o/sec, change the equation to represent the motion of the cylinder in terms of time.
c) For the equation, describe how the various factors (a, k, c, and d) relate to the motion the cylinder.
d) Suppose the train started out with the cylinder at the far right end. How far along the track does the train travel before the cylinder reaches the original position?
e) What position (in terms of angle) would the point on the cylinder be in after the train traveled 200 cm?
f) What would the effect of introducing a larger wheel be on the equation?
g) What would the effect of introducing a longer connecting rod be on the equation (assume the wheel stays in the same location)?
a) To create an equation modeling the motion of the cylinder with respect to the rotational angle of a point on the wheel, we should first collect data points by rotating the point and recording the corresponding positions of the cylinder. Then, we can use this data to find the values for the parameters a, k, c, and d in the equation y = a sin k(θ - c) + d.
b) To represent the motion of the cylinder in terms of time, we need to relate the rotational angle θ to time. Since the speed of rotation is given as 150o/sec, we can use the equation θ = 150t, where t represents time in seconds. Now, we can substitute this equation into the original equation y = a sin k(θ - c) + d to represent the motion of the cylinder with respect to time.
c) In the equation y = a sin k(θ - c) + d:
- Parameter a represents the amplitude, which determines the maximum displacement of the cylinder from its mean position.
- Parameter k represents the period of oscillation or frequency, which determines how quickly the cylinder goes through one complete cycle of motion.
- Parameter c represents the phase shift, which determines the initial position of the cylinder when θ is zero.
- Parameter d represents the vertical shift, which determines the mean position of the cylinder.
d) If the cylinder started at the far right end and needs to reach the original position, we need to find the value of θ where y = 0 in the equation y = a sin k(θ - c) + d. This value corresponds to one complete cycle or period of the motion. By solving for θ = 2π or 360 degrees, we can determine the amount of rotation required for the cylinder to reach its original position.
e) To find the position (in terms of angle) of the point on the cylinder after the train traveled 200 cm, we need to determine the corresponding value of θ in the equation y = a sin k(θ - c) + d. We can use the given information to relate the distance traveled to the rotational angle and solve for θ.
f) Introducing a larger wheel would have an effect on the equation by changing the value of k, the period or frequency. A larger wheel would cover a greater distance per rotation, resulting in a larger value of k and a shorter period of oscillation.
g) Introducing a longer connecting rod would not have a direct effect on the equation, assuming the wheel stays in the same location. The motion of the cylinder is determined by the rotational motion of the wheel, and the connecting rod acts as a linkage between the two. As long as the point on the wheel that is being tracked remains in the same location, the length of the connecting rod does not affect the equation directly.
a) To create an equation in the form y = a sin k(θ - c) + d, we need to collect data by rotating the point on the wheel and noting the corresponding positions of the cylinder.
b) If the speed of rotation is 150o/sec, we can use the formula θ = ωt, where θ is the angle in radians, ω is the angular speed in radians per second, and t is the time in seconds. Therefore, the equation representing the motion of the cylinder in terms of time would be y = a sin k(ωt - c) + d.
c) The various factors in the equation relate to the motion of the cylinder as follows:
- a represents the amplitude of the motion, which determines how far the cylinder moves vertically from its equilibrium position.
- k controls the frequency or rate at which the cylinder oscillates. A higher value of k leads to more rapid oscillations.
- c represents a phase shift, determining where in the cycle the motion starts. A positive value of c shifts the motion to the right, while a negative value shifts it to the left.
- d is a vertical shift of the entire equation, determining the average position of the cylinder.
d) To determine how far the train travels before the cylinder reaches the original position, you need to calculate the period of the motion. The period can be found using T = 2π/k, where T is the period and k is the frequency. Once you have the period, multiply it by the speed of rotation in order to find the distance traveled.
e) To find the position (angle) of the point on the cylinder after the train has traveled 200 cm, you need to find the corresponding value of θ in the equation y = a sin k(θ - c) + d. Rearrange the equation to solve for θ, substitute the known values of a, k, c, and d, and solve for θ. This will give you the position in terms of angle.
f) If a larger wheel is introduced, it would affect the equation by changing the amplitude (a) and frequency (k) of the motion. The larger the wheel, the larger the amplitude of the motion, resulting in a greater vertical displacement. The frequency of the motion may also change, depending on the design and characteristics of the new wheel.
g) Introducing a longer connecting rod while keeping the wheel in the same location would affect the equation by changing the amplitude (a) and the phase shift (c) of the motion. A longer connecting rod will result in a larger amplitude of the motion, similar to the effect of a larger wheel. The phase shift will also change, as the longer rod will cause the motion to start at a different point in the cycle.