A 2.25×10 4 N elevator is to be accelerated upward by connecting it to a counter

Question

A 2.25×104 N elevator is to be accelerated upward by connecting it to a counterweight using a light (but strong!) cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875 kg and it is 1.50 m in diameter.

A)How heavy should the counterweight be so that it will accelerate the elevator upward through 6.75 m in the first 4.00 s , starting from rest?

M= ____Kg

B) Under these conditions, what is the tension in the cable on elevator side of the pulley?

T= ____N

C)Under these conditions, what is the tension in the cable on counterweight side of the pulley?

T= ____N

Please show work, and feel free to explain the conceptual aspect .

Explanation / Answer

Given that

The weight of the elevator is (w) =2.25×104N

Then the mass of the elevator is (m1) =2293.577kg

The mass of the pulley is (M) =875kg

The diamter of the pulley (d) =1.50m

The radiu of the pulley (R) =0.75m

The elevator reaches to a height (h) =6.75m and time t =4.00s

The acceleration due to gravity (g) =9.81m/s2

Then the acceleration of the elevator is given by

s =(1/2)at2

h =(1/2)at2

Then a =2h/t2 =0.84m/s2

Now the elevator which is accelerating upward is given by

ma1 =T1-m1g

Then T1 =m1g+m1a

now for the weight

m2a =m2g-T2

T2 =m2g-m2a

And now for the pulley is

(T2-T1)R =I*alpha

(m2g-m1a-m1g-m1a)R =(1/2)MR2 (a/R)

Now the mass of the counter weight is given by

m2 =(0.5)Ma+m1g+m1a/g-a =0.5*(875)*(0.84)+(2293.577kg)(9.81)+(2293.577kg)(0.84)/(9.81-0.84)

                                          =367.5+22500+1926.604/8.97

                                          =24794.1046/8.97

                                          =2764.114kg

b)

The tension in the cable is given by T1 =m1g+m1a =m1(g+a) =2293.577kg(9.81+0.84) =24426.595N

c)

The tension in the cable on counterweight side of the pulley is given by

                T2 =m2g-m2a =m2 (g-a) =(2764.114kg)(9.81-0.84) =24794.1025N

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