3)-A 9-kg block is held by a pulley system shown in fig. What force must the person exert in the following cases: a) to hold the block at rest. b) to lower it at 2m/s. c) to raise it with an acceleration of 0.5 m/s2? Ignore the mass of small pulley.
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.
To solve this problem, we first need to understand the forces involved in the pulley system. In a pulley system like the one shown in the figure, there are two main forces to consider: the tension force in the rope and the force exerted by the person holding or manipulating the rope.
a) In the case of holding the block at rest, the force exerted by the person must be equal to the weight of the block. The weight of an object can be calculated by multiplying its mass by the acceleration due to gravity, which is approximately 9.8 m/s^2.
So, the force the person must exert to hold the block at rest is:
Force = mass * acceleration due to gravity
Force = 9 kg * 9.8 m/s^2
Force = 88.2 N
Therefore, the person must exert a force of 88.2 Newtons to hold the block at rest.
b) To lower the block at a constant speed of 2 m/s, the force exerted by the person must be equal to the tension force in the rope. Since the block is moving at a constant speed, the net force acting on it is zero.
The tension force in the rope can be found by using Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration:
Net force = mass * acceleration
Since the block is moving at a constant speed, the acceleration is zero. Therefore, the net force is also zero, which means the tension force in the rope is equal to the weight of the block.
So, to lower the block at 2 m/s, the person must exert a force equal to the weight of the block, which we calculated to be 88.2 N in part (a).
c) To raise the block with an acceleration of 0.5 m/s^2, the person must exert a force greater than the weight of the block. This is because the block requires an additional force to overcome its weight and provide the desired acceleration.
The force required to raise the block with the given acceleration can be calculated using Newton's second law:
Net force = mass * acceleration
The net force consists of the tension force in the rope (upwards) and the weight of the block (downwards). Therefore, the equation becomes:
Tension force - Weight = mass * acceleration
We know the weight of the block is 88.2 N, its mass is 9 kg, and the acceleration is 0.5 m/s^2. Plugging in these values, we can solve for the tension force:
Tension force - 88.2 N = 9 kg * 0.5 m/s^2
Tension force - 88.2 N = 4.5 N
Tension force = 92.7 N
So, to raise the block with an acceleration of 0.5 m/s^2, the person must exert a force of 92.7 Newtons.