A block having mass 90 kg.just contact the top of the free end of spring.
(1) find the maximum deflection of spring if the block is released instantaneously with zero initial velocity.
(2) if the block is very gradually lowered on the spring.
To find the maximum deflection of the spring in both situations, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Let's consider the situation in both cases separately:
(1) When the block is released instantaneously with zero initial velocity:
In this scenario, the block will experience a sudden force due to gravity acting on it. The weight of the block can be calculated using the formula:
Weight = mass × acceleration due to gravity = 90 kg × 9.8 m/s² = 882 N.
Since the block is released from rest, it will initially compress the spring until it reaches its maximum deflection. At this point, the upward force exerted by the spring (F_spring) will balance the weight of the block, causing it to stop accelerating downwards. This maximum deflection can be calculated by applying Hooke's Law:
F_spring = Weight
k × maximum deflection = m × g
k × maximum deflection = 882 N (since m × g = 882 N)
Here, 'k' represents the spring constant, which is a measure of the stiffness of the spring and is usually provided. Once you know the spring constant, you can rearrange the equation and solve for the maximum deflection (x):
maximum deflection = Weight / k
(2) If the block is very gradually lowered on the spring:
In this scenario, as the block is gradually lowered onto the spring, it will compress the spring gradually until it reaches its equilibrium position where the force exerted by the spring equals the weight of the block.
Since it is lowered gradually, we consider the block to be in static equilibrium when it reaches its maximum deflection. At this point, the force exerted by the spring (F_spring) is equal and opposite to the weight of the block (Weight) to maintain equilibrium.
Using Hooke's Law:
F_spring = Weight
k × maximum deflection = m × g
k × maximum deflection = 882 N
As mentioned earlier, you will need the spring constant (k) to calculate the maximum deflection (x).
Note: In both cases, make sure the units are consistent (e.g., N/m for the spring constant, kg for mass, m/s² for acceleration due to gravity) to get the correct numerical value for the maximum deflection.