Question

KC has a piece of gum stuck to her bike tire, which applies a forward torque (i.e. a force to roll forwards) on the tire's movement. The torque varies in a periodic way that can be modeled approximately by a trigonometric function. When the gum is on the front of the tire, its weight is pulling forwards with a maximum torque of 0.01 Nm (Newton metre, the SI unit for torque), and when it's on the back of the tire, it's pulling backwards with a minimum torque of −0.01 Nm. The maximum torque is reached once in every rotation, which is every 1.2 meters. The first time it reaches its maximum torque is 0.3 meters into the race. Find the formula of the trigonometric function that models the torque τ the gum applies on the tire d meters into the race. Define the function using radians.

Answer 1

The trigonometric function modeling the torque τ is τ = 0.01 cos(π/0.6 × (d - 0.3)), with the amplitude of 0.01 Nm, frequency π/0.6 because of the 1.2-meter period, and a phase shift of 0.3 meters to account for the starting point of the maximum torque.

Step-by-step explanation:

To find the formula of the trigonometric function that models the torque τ applied by the gum on the tire d meters into the race, we consider the given properties of the function. The torque is a periodic function with a period equivalent to the circumference of the tire, which is 1.2 meters. Since the torque is maximum at 0.3 meters and the max torque is 0.01 Nm, we can use a cosine function that starts at its maximum value and has a period of 1.2 meters.

The general form of a cosine function is τ = A cos(B(d - C)) + D, where A is the amplitude, B is the frequency, C is the horizontal shift, and D is the vertical shift. Given the maximum torque of 0.01 Nm and a minimum torque of -0.01 Nm, the amplitude A is 0.01 Nm. Since the maximum occurs every 1.2 meters, which corresponds to a full cycle of 2π radians, the frequency B is π/0.6. The horizontal shift C is 0.3 meters, as the first maximum of the torque occurs at this point. The vertical shift D is zero since the function oscillates symmetrically above and below the horizontal axis with no vertical displacement.

Therefore, the equation for the torque τ in terms of the distance d is:

τ = 0.01 cos(π/0.6 × (d - 0.3))