(a) An object is thrown up a smooth plane inclined at an angle of 30° to the horizontal. If the plane is 15m long and the object comes to rest at the top, find the :
(i) initial speed of the object ; (ii) time taken to reach the top.
(b)
Force of magnitudes (5 N, 5sqrt{3} N, 10 N, 5sqrt{3} N) and (5 N) act on a body P, of mass 5 kg as shown in the diagram. Find the :
(i) magnitude of the resultant force ; (ii) acceleration of the body.
Explanation
(a)
If d = deceleration, then retarding force = (md).
This is also the force down the plane (mg sin theta)
(therefore md = mgsin theta implies d = g sin theta)
= (10 sin 30° = 10 times frac{1}{2} = 5 ms^{-2})
(i) (v^{2} = u^{2} + 2as)
At the highest point, v = 0 ms(^{-1}).
(0^{2} = u^{2} – 2(5)(15) )
(u^{2} = 150 implies u = sqrt{150} = 12.247 ms^{-1})
(ii) (v = u – dt)
(0 = 12.247 – 5t implies 5t = 12.247)
(t = frac{12.247}{5} = 2.4494 approxeq 2.45 s)
(b)(i) Resultant force
(F = begin{pmatrix} 10 cos 0° \ 10 sin 0° end{pmatrix} + begin{pmatrix} 5sqrt{3} cos 30° \ 5sqrt{3} sin 30° end{pmatrix} + begin{pmatrix} 5 cos 30° \ 5 sin 30° end{pmatrix} + begin{pmatrix} 5 cos 300° \ 5 sin 300° end{pmatrix} + begin{pmatrix} 5sqrt{3} cos 330° \ 5 sqrt{3} sin 330° end{pmatrix})
= (begin{pmatrix} 10 times 1 \ 10 times 0 end{pmatrix} + begin{pmatrix} 5sqrt{3} times frac{sqrt{3}}{2} \ 5 sqrt{3} times frac{1}{2} end{pmatrix} + begin{pmatrix} 5 times frac{1}{2} \ 5 times frac{sqrt{3}}{2} end{pmatrix} + begin{pmatrix} 5 times frac{1}{2} \ -5 times frac{sqrt{3}}{2} end{pmatrix} + begin{pmatrix} 5sqrt{3} times frac{sqrt{3}}{2} \ -5sqrt{3} times frac{1}{2} end{pmatrix})
= (begin{pmatrix} 10 \ 0 end{pmatrix} + begin{pmatrix} 7.5 \ 2.5 sqrt{3} end{pmatrix} + begin{pmatrix} 2.5 \ 2.5sqrt{3} end{pmatrix} + begin{pmatrix} 2.5 \ -2.5 sqrt{3} end{pmatrix} + begin{pmatrix} 7.5 \ -2.5 sqrt{3} end{pmatrix})
= (begin{pmatrix} 30 \ 0 end{pmatrix})
Magnitude of resultant = (sqrt{30^{2}} = 30N)
(ii) (F = ma implies a = frac{F}{m})
= (frac{30}{5} = 6 ms^{-2})