A quarry crane is used to lift massive rocks from a quarry pit. Consider the sim

Question

A quarry crane is used to lift massive rocks from a quarry pit. Consider the simplified model of such a crane shown in the figure. (Figure 1) The ends of two poles are anchored to the ground at the same point (point O). From this point, one pole rises vertically and the second pole rises at an angle. The vertical pole has its free end connected to the ground via an unstretchable, massless rope labeled rope 1. A second rope, labeled rope 2, connects the free ends of the two poles. The angle between the tilted pole and rope 2 is . Both poles have length L and can be considered massless for the purposes of this problem. Hanging from the end of the second pole, via rope 3, is a granite block of mass m. Throughout this problem use g for the magnitude of the acceleration due to gravity.

Part C:

Find T1, the tension in rope 1.

Express the tension in terms of some (or all) of the following quantities: m, g, L, ?, and ?.

Explanation / Answer

a) The tension T_3 is just the weight of the block, providing the block is at rest or moving up at a constant speed.

T_3 = mg

b) For this what I did was first to determine the angle between the two poles. Since the poles are of equal length, rope 2 makes the same angle with both poles ie ?.

So the angle between the two poles is ?-2?.

This angle is the same as the angle between rope 1 and the tilted pole.

Now we can find the torque that rope 1 applies about point O.

To find the torque we need the force rope 1 applies to the end of the tilted pole, in the direction perpendicular to the length of the pole in the drawing. This makes an angle of ?/2 - (?-2?) = 2? - ?/2 to rope 1.

So the torque applied by rope 1 about point O is

?(1) = Lmgcos(2?-?/2) = Lmgsin(2?)

Now we can find the torque applied about point O by rope 2

This torque ?(2) must equal ?(1) so that there is no movement in the tilted pole.

Using the same method as for finding ?(1), noting that the angle between rope 2 and the tilted pole is ? and using T as the tension in rope 2 you should be able to show that

?(2) = LTsin(?)

Now ?(2) = ?(1)

LTsin(?) = Lmgsin(2?)

Tsin(?) = mgsin(2?) = 2mgsin(?)cos(?) (using a trig identity)

T = 2mgcos(?)

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