I know the answers are t=1.1049s and v= 7.91 ft/s, but i can\'t firgure out how
ID: 1717950 • Letter: I
Question
I know the answers are t=1.1049s and v= 7.91 ft/s, but i can't firgure out how to get to them. I think I am making a small mistake somewhere.
12.40 The horizontal force P8- 2t lb (t is the time measured in seconds) is applied to the 5-lb collar that slides along the inclined rod. At time t = 0, the position coordinate of the collar is x = 0, and its velocity is 0-10 ft/s directed down the rod. Find the time and the speed of the collar when it returns to the position 0 for the first time. Neglect friction 5 1b Vo Fig. P12.40Explanation / Answer
>> Force, P = 8 - 2t {lb}
As, along the motion of collar, or along inclined plane,
Force Acting = P*(4/5)= 0.8*P = 1 - 1.6t
As, Force = mass*Acceleration
=> Acceleration, A = Force/<ass = (1 - 1.6t)/5 = 0.2 - 0.32t
As, A = dV/dt [ V = Velocity ]
=> dV = (0.2 - 0.32t) dt
>> Integrating both sides,
=> V = 0.2t - 0.16t2 + C [ C = IntegrationConstant ]
>> As, V = dX/dt , X= position
=> dX = (0.2t - 0.16t2 + C) dt
Integrating both sides,
=> X = 0.1t2 - (0.16/3)t3 + Ct + D [ D = Integration Constant ]
>> Now, as at t = 0, x = 0
=> D = 0,
Also, at t = 0, V = 10 ft/s,
=> C = 10
=> V = 0.2t - 0.16t2 + 10
and,
X = 0.1t2 - (0.16/3)t3 + 10*t
>> Now, for again to X = 0
=> 0.1t2 - (0.16/3)t3 + 10*t = 0
=> t = 14.6626 sec
=> Required Time = 14.66 sec ............ANSWER............
>> At this point, Velocity , V
V = 0.2*14.66 - 0.16*14.662 + 10 = - 21.47 ft/sec
=> Required Velocity of collar = 21.47 ft/sec .......ANSWER....