Biology Forums - Study Force

A DC electric motor is connected to a wheel via a gearbox. The aim is to find when to engage
the gear so that the torque at the wheel is always the largest possible.
On its own, the motor generates a torque 𝑀 (Ω) (in Nm) that depends on the rotation speed Ω
(in rpm) of the motor shaft:
𝑀 (Ω) = 0.3 (1 − Ω/400)

The torque 𝑇 (𝜔) exerted at the wheel depends on the rotation speed 𝜔 of the wheel, and on
whether the gear is engaged:
𝑇0 (𝜔) = 𝑀 (𝜔), if the gear is not engaged,
𝑇1 (𝜔) = 1/3 𝑀  𝜔/3
, if the gear is engaged.
(a) Write explicit expressions for 𝑇0 (𝜔) and 𝑇1 (𝜔) as functions of 𝜔 given the definition of
𝑀 (Ω).
(b) Plot the two functions you found in (a) on a same graph for positive values of 𝜔.
(c) Find the optimal gear shifting point 𝜔∗ that ensures that the torque 𝑇 (𝜔) is always as
large as possible.
Hint: Look for a graphical solution first by using the graph from Part (b).

Hi Joseph n (https://biology-forums.com/index.php?action=profile;u=0)

I found a similar question posted elsewhere that has the same three questions answered but with different values. See if it helps... If not, reply back and I'll provide a more custom response.

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_28_12.png)

Answers:

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_28_33.png)

b. The graph of the two functions are:

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_28_43.png)

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_29_20.png)

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_29_25.png)

c.

(https://biology-forums.com/gallery/qpics/6_29_02_24_8_30_06.png)