An elevator (mass 4500 kg ) is to be designed so that the maximum acceleration is 0.0680 g . what is the maximum force the motor should exert on the supporting cable?

For this case what you should do is to start from the definition that force is equal to mass by acceleration.
 We have then:
 f = m * a
 where,
 m = 4500Kg
 a = 0.0680g
 g = 9.81 m / s ^ 2
 Substituting the values:
 f = (4500) * ((0.0680) * (9.81))
 f = 3001.86 N
 
 the maximum force the motor should exert on the supporting cable is
 f = 3001.86 N

More information

As newton's first law states a object in motion stays in motion