Question

derive the three equation of rotational motion under constant acceleration from first principle​

Answer 1

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Analogous to Newton's law (F = Δ( mv)/Δ t) there is a rotational counterpart for rotational motion: t = Δ L/Δ t, or torque is the rate of change of angular momentum.

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Answer 2

Answer:

angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.

To begin, we note that if the system is rotating under a constant acceleration, then the average angular velocity follows a simple relation because the angular velocity is increasing linearly with time. The average angular velocity is just half the sum of the initial and final values:

ω¯=ω0+ωf2.(10.3.1)

From the definition of the average angular velocity, we can find an equation that relates the angular position, average angular velocity, and time:

ω¯=ΔθΔt.(10.3.2)

Solving for θ , we have

θf=θ0+ω¯t,(10.3.3)

where we have set t0 = 0. This equation can be very useful if we know the average angular velocity of the system. Then we could find the angular displacement over a given time period. Next, we find an equation relating ω , α , and t. To determine this equation, we start with the definition of angular acceleration:

α=dωdt.(10.3.4)

We rearrange this to get α dt = d ω and then we integrate both sides of this equation from initial values to final values, that is, from t0 to t and ω0 to ωf . In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals:

α∫tt0dt′=∫ωfω0dω.(10.3.5)

Setting t0 = 0, we have

αt=ωf−ω0.(10.3.6)

We rearrange this to obtain

ωf=ω0+αt,(10.3.7)

where ω0 is the initial angular velocity. Equation 10.3.7 is the rotational counterpart to the linear kinematics equation vf = v0 + at. With Equation 10.3.7 , we can find the angular velocity of an object at any specified time t given the initial angular velocity and the angular acceleration.