Problem 3:

A student is carrying an 8 kg mass on a 3 m long shovel, extended horizontally. The mass of the shovel is negligible. The student uses both hands, one located at the end of the shovel, and the other located 0.75 m from the end (See diagram).

a) What are the forces exerted by each of her hands if the mass is to remain balanced?
b) What is the direction of the torque on the 8 kg mass (about an axis through her left hand)?
c) She now lets go with the hand that was on end of the shovel (her right hand). What is the initial angular acceleration of the 8 kg mass?

Solution:

a) Static equilibrium problem. Sum of the forces and sum of the torques must be zero for equilibrium. Call the force exerted by her left hand F_L and the force exerted by her right hand F_R. Choose (as a guess) both of these forces acting upwards. We have from the force equation

     F_L + F_R - Mg = 0

where M = 8 kg.

Consider torques about her left hand. We have

     F_R(0.75 m) + Mg(2.25 m) = 0

solving this yields

    F_R = - (8 kg)(9.8 m/s²)(2.25m)/(0.75 m)  =  - 235 N

i.e. the force exerted by her right hand is negative (downward direction)

Then using the force equation above, we get

    F_L = Mg - F_R  =  (8 kg)(9.8 m/s²) + 235 N  =  313 N

i.e. the force exerted by her left is in the upward direction.

b) The torque on the 8 kg mass (due to gravity) is into the page (using the right hand rule) - it wants to create a motion in the clockwise sense.

c) torque  = = I  so the angular acceleration    is

   = /I = (M g R)/(MR²) = Mg/MR = g/R = (9.8 m/s²)/(2.25 m) = 4.36 m/s²

This is the magnitude of the initial angular acceleration; its direction is the direction of the torque, i.e. into the page.

Problem 4
Test 3
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