2. At time t =0, the current to the dc motor is reversed, resulting in an angular displacement of the motor shaft given by θ = (198 rad/s)t – (24 rad/s2)t2 – (2 rad/s3)t3
a. At what time is the angular velocity of the motor shaft zero?
b. Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity.
c. Through how many revolutions does the motor shaft turn between the time when the current is reversed and the instant when the angular velocity is zero?
d. How fast was the motor shaft rotating at t=0, when the current was reversed?
e. Calculate the average angular velocity for the time period from t = 0 to the time calculated in part (a).
For question (c) you can not use the kinematic equation because the acceleration of the system is 'not constant'. The more correct method would be:
Number of Revolutions = 1rev/(2*pi radians) x [ (Angular displacement at 3 sec) - (Angular displacement at 0 sec) ]
φ =198t-24t²-2t³
(a) ω=dφ/dt=198 -48t-6t²,
ω=0 => 198 -48t-6t²=0,
6t²+48t-198=0,
t²+8t – 33 =0,
t= -4±√(16+33 ) =-4±7,
t=-11 (impossible),
t=3 s.
(b) ε=dω/dt= - 48 -12t.
t=3 => ε=-48 -12•3 = - 84 rad/s².
(c) t=0, ω= 198 -48t-6t² =>
ω₀= 198 rad/s,
ε= - 84 rad/s²,
φ = ω₀t+εt²/2=
=198t-84•t²/2=
=198•3 - 84•9/2=216 rad.
φ = 2πN =>
N= φ/2π=216/2π=34.4 rev.
(d) ω₀= 198 rad/s.
(e) average angular velocity
ω(ave) = φ/t=216/3= 72 rad/s.
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a. The angular velocity of the motor shaft will be zero when the rate of change of angular displacement with respect to time (angular velocity) is zero. So, we need to find the time when (dθ/dt) = 0.
To find the time when the angular velocity is zero, we need to differentiate the equation for angular displacement with respect to time:
dθ/dt = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
Setting dθ/dt = 0:
0 = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
We can solve this equation to find the time when the angular velocity is zero.
b. To calculate the angular acceleration at the instant when the motor shaft has zero angular velocity, we need to differentiate the equation for angular velocity with respect to time:
d(θ')/dt = -2(24 rad/s^2) - 6(2 rad/s^3)t
Substituting t = 0 (since we want the instant when angular velocity is zero), we can find the angular acceleration.
c. To calculate the number of revolutions the motor shaft turns between the time when the current is reversed and the instant when the angular velocity is zero, we need to integrate the angular velocity equation with respect to time over the given time range. The number of revolutions can be calculated by dividing the total angular displacement by 2π.
d. To find how fast the motor shaft was rotating at t = 0, when the current was reversed, we need to substitute t = 0 into the equation for angular velocity.
θ' = (198 rad/s) - 2(24 rad/s^2)(0) - 3(2 rad/s^3)(0)^2
e. To calculate the average angular velocity for the time period from t = 0 to the time when the angular velocity is zero, we need to find the total change in angular displacement during that time period and divide it by the total time taken.
To answer these questions, we need to consider the equations provided for the angular displacement θ as a function of time t.
a. To find the time at which the angular velocity is zero, we need to look for the value of t for which the angular displacement rate is zero. In other words, we need to find the time t when dθ/dt = 0.
Given that θ = (198 rad/s)t – (24 rad/s^2)t^2 – (2 rad/s^3)t^3, we can differentiate θ with respect to t to find dθ/dt:
dθ/dt = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
Now, let's set dθ/dt = 0:
0 = (198 rad/s) - 2(24 rad/s^2)t - 3(2 rad/s^3)t^2
This equation represents a quadratic equation in t. We can solve it using the quadratic formula. The time at which the angular velocity is zero will be the value of t that satisfies this equation.
b. To calculate the angular acceleration at the instant when the motor shaft has zero angular velocity, we need to find the second derivative of θ with respect to t. In other words, we need to find d^2θ/dt^2. We can differentiate the equation for dθ/dt above:
d^2θ/dt^2 = -2(24 rad/s^2) - 6(2 rad/s^3)t
Substitute the value of t obtained in part (a) into this equation to calculate the angular acceleration at that instant.
c. To find the number of revolutions the motor shaft turns between the time when the current is reversed and the instant when the angular velocity is zero, we need to calculate the total change in angular displacement during that time interval. We can do this by integrating the equation for angular displacement with respect to t from t = 0 (when the current is reversed) to the time obtained in part (a).
d. To calculate the angular velocity of the motor shaft at t = 0, we can substitute t = 0 into the equation for dθ/dt obtained in part (a).
e. To calculate the average angular velocity for the time period from t = 0 to the time obtained in part (a), we need to find the total change in angular displacement during that time interval and divide it by the duration of the interval. We can do this by calculating the integral of dθ/dt with respect to t from t = 0 to the time obtained in part (a), and then dividing it by the duration of the interval.