To solve this problem, we need to consider the forces acting on the block in each case.
a) To hold the block at rest:
In this case, the person needs to exert a force equal in magnitude but opposite in direction to the force of gravity acting on the block. The force of gravity can be calculated using the formula F = mg, where m is the mass of the block (9 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). So, the force the person must exert is:
F = (mass of the block) × (acceleration due to gravity)
F = 9 kg × 9.8 m/s^2
F = 88.2 N
Therefore, the person must exert a force of 88.2 N to hold the block at rest.
b) To lower it at 2 m/s:
In this case, the person must exert a force that balances the force of gravity and provides an additional force to accelerate the block downward. The net force required can be calculated using the equation F = ma, where m is the mass of the block (9 kg) and a is the acceleration (2 m/s^2) at which the block is being lowered. The force the person must exert is:
F = (mass of the block) × (acceleration due to gravity + acceleration of lowering)
F = 9 kg × (9.8 m/s^2 + 2 m/s^2)
F = 9 kg × 11.8 m/s^2
F = 106.2 N
Therefore, the person must exert a force of 106.2 N to lower the block at 2 m/s.
c) To raise it with an acceleration of 0.5 m/s^2:
In this case, the person must exert a force that balances the force of gravity and provides an additional force to accelerate the block upward. The net force required can be calculated using the equation F = ma, where m is the mass of the block (9 kg) and a is the acceleration (0.5 m/s^2) at which the block is being raised. The force the person must exert is:
F = (mass of the block) × (acceleration due to gravity - acceleration of raising)
F = 9 kg × (9.8 m/s^2 - 0.5 m/s^2)
F = 9 kg × 9.3 m/s^2
F = 83.7 N
Therefore, the person must exert a force of 83.7 N to raise the block with an acceleration of 0.5 m/s^2.