Complete Question
The complete question is shown on the first uploaded image
Answer:
The value of t is [tex]t = 15.08 \ s[/tex]
Explanation:
From the question we are told that the angular speed is
The initial angular speed [tex]w_f = 150 rev/min = \frac{2 \pi }{60s} * 150 = 15.71rad /s[/tex]
The force is [tex]T = 20 N[/tex]
The radius is [tex]r = 400mm = \frac{400}{1000} = 0.4m[/tex]
The total mass of the four sphere is [tex]m_a = (4*3) = 12kg[/tex]
The initial velocity is [tex]v_i = 0m/s[/tex]
The radius of the pully is [tex]r_p = 100mm = \frac{100}{1000} = 0.10m[/tex]
The initial time is [tex]t_1 = 0s[/tex]
The final time is [tex]t = t[/tex]
Generally the final velocity of the sphere is mathematically represented as
[tex]v_f = r w_f[/tex]
[tex]v_f=15.7 r[/tex]
The angular impulse momentum principle can be represented methematically as
[tex](H_O)_i + \int\limits^{t_1}_{t_2}{(T \cdot r_{p})} \, dt = (H_O)_f[/tex]
[tex]r(m_a v_i ) + \int\limits^{t_2}_{t_1} {T \cdot r_p} \, dt = r(m_a v_f)[/tex]
[tex]r(m_a v_1 ) + T \cdot r_p (t_{2} -t_{1}) = r( m_a * 15.7 r )[/tex]
Substituting values
[tex]0.4 * (12* 0) + (20 *0.10 * (t-0)) = 0.40 * (12* 0.40 *15.7)[/tex]
=> [tex]0 + 2t = 30.16[/tex]
=> [tex]t = 15.08 \ s[/tex]
