Answer:
(a). The acceleration of the block is 1.793m/s.
(b). The string has a tension of 2.8N.
Explanation:
The forces acting on the block are the tension [tex]T[/tex] in the string, the frictional force [tex]F_k[/tex], and the gravitational force [tex]mg\: sin(\theta)[/tex], together they induce acceleration [tex]a[/tex] in the block; therefore,
[tex](1). \: \: ma= mg\:sin(\theta)- T-F_k[/tex].
Now, the frictional force [tex]F_k[/tex] is
[tex](2). \: \: F_k = \mu N =\mu \: mg\: cos(\theta)[/tex]
and the tension [tex]T[/tex] in the string is what causes the angular acceleration [tex]\alpha[/tex] in the flywheel; therefore,
[tex]I\alpha = TR[/tex]
[tex](3). \: \: \alpha = \dfrac{TR}{I},[/tex]
where [tex]R[/tex] is the distance from the axis of the flywheel and [tex]I[/tex] is its moment of inertia.
(a).
Since the linear acceleration of the flywheel is [tex]a = \alpha R[/tex], equation (3) gives
[tex]a = \dfrac{TR^2}{I}[/tex]
which when solved for [tex]T[/tex] gives
[tex](4).\: \: T = \dfrac{aI}{R^2}[/tex]
putting this and equation (2) into equation (1) gives
[tex]ma= mg\:sin(\theta)- \dfrac{aI}{R^2} -\mu mg\: cos(\theta)[/tex]
solving for [tex]a[/tex] we get:
[tex]ma+ \dfrac{aI}{R^2}= mg[\:sin(\theta)-\mu cos(\theta)][/tex]
[tex]a(m+ \dfrac{I}{R^2})= mg[\:sin(\theta)-\mu cos(\theta)][/tex]
[tex]\boxed{a = \dfrac{mg[\:sin(\theta)-\mu cos(\theta)]}{(m+ \dfrac{I}{R^2})}.}[/tex]
Putting in [tex]m=5kg[/tex], [tex]g= 9.8m/s^2[/tex], [tex]\theta = 30^o[/tex],[tex]\mu = 0.3[/tex], [tex]I = 0.25kg\:m^2[/tex], and [tex]R = 0.4m[/tex] we get:
[tex]a = \dfrac{(5kg)(9.8m/s^2)[\:sin(30^o)-(0.3) cos(30)]}{(5kg+ \dfrac{0.25kg\:m^2}{(0.4m)^2})}[/tex]
[tex]\boxed{a= 1.793m/s.}[/tex]
Thus, the acceleration of the block is 1.793m/s.
(b).
The tension [tex]T[/tex] in the string is given by the equation (4)
[tex]T = \dfrac{aI}{R^2}[/tex]
putting in [tex]I = 0.25kg\:m^2[/tex], [tex]R = 0.4m[/tex], and [tex]a =1.793m/s[/tex] we get:
[tex]T = \dfrac{(1.793m/s^2)(0.25 kg\; m^2)}{(0.4m)^2}[/tex]
[tex]\boxed{T = 2.8N.}[/tex]
Thus, the string has a tension of 2.8N.
