Question

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)=γt+βt3, where γ= 0.350 rad/s and β= 1.20×10^−2 rad/s^3.
a. Calculate the angular velocity of the merry-go-round as a function of time. Express your answer in radians per second in terms of γ, β, and t. ω(t) = rad/sec
b. What is the initial value ω0 of the angular velocity? Express your answer in radians per second. ω0 = rad/s
c. Calculate the instantaneous value of the angular velocity ω(t) at time t=5.00s. Express your answer in radians per second. ω(5.00) = rad/s
d. Calculate the average angular velocity ωav for the time interval t=0 to t=5.00 seconds. Express your answer in radians per second. ωav = rad/s

Answer 1

Answer:

$a. \ \gamma +3\beta t^2\\\\b. \ 0.350rad/s\\\\\\c. \omega_i_n_s_t=1.25rad/s\\\\d. w_a_v=0.7rad/s$

Explanation:

a. Angular velocity,$\omega$ is defined as the rate of change of position of a rotating body with respect to time. so omega,$\omega$= theta/time and theta =the position angle.

-Angular velocity of the merry-go-round can thus be calculated and  expressed as:

$\omega=\frac{d\thata}{dt}=\frac {d(\gamma t+\beta t^3)}{dt}\\=\gamma +3\beta t^3$

b. From a above, we know that our angular velocity is $\omega=\gamma +3\beta t^3$ ,we can substitute $t=0$ to find our initial velocity.

$\omega_t_=_o=\gamma+3\beta \times (0)^3\\w_t_=_0=0.350rad/s$

Hence the value of angular velocity at t=0 is 0.350rad/s

c. To calculate the instantaneous angular velocity at t=5.00s, we

$\omega_t_=_5=\gamma +3\beta t^3\\=0.350+ 3\times 0.0120 \times 5^3\\=1.25rad/s$

Therefore, the instanteneous angular velocity is 1.25rad/s

d. The average angular velocity between t=0 and 2=5s is calculated as

-At t=0,$(\theta)=\gamma \times 0+\beta \times 0=0$

-At time t=5s, $\omega=5\gamma +125\beta\times=3.5rad$

Therefore, the average angular velocity is calculated as

$\omega_a_v=\frac{\bigtriangleup \theta}{\bigtriangleup t}=\frac{\theta_2-\theta_1}{t_2-t_1}\\=\frac{3.5-0}{5-0}\\\\=0.7rad/s$

Hence the average angular velocity is 0.7rad/s

#instantenous angular velocity does not increase linearly .