a plank of mass 5kg is placed on a frictionless horizontal table. further a block of mass 2kg is placed over the plank. A mass-less spring of natural length 2m is fixed to the plank by its one end. The other end of spring is compressed by the block by half of its springs natural length. The system is now released from the rest. What is the velocity of the plank when block leaves the plank?
To determine the velocity of the plank when the block leaves the plank, we can apply the principles of conservation of energy.
First, let's consider the potential energy of the spring when it is compressed. The spring potential energy is given by the formula:
PE_spring = (1/2)kx^2
Where k is the spring constant and x is the displacement from the natural length. Given that the spring's natural length is 2m and it is compressed by half of its natural length, x = 1m.
Next, we need to find the spring constant. The spring constant can be determined using Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position. Mathematically, it can be written as:
F = -kx
Where F is the force exerted by the spring and x is the displacement from the equilibrium position. In this case, when the block leaves the plank, the displacement of the spring is 1m.
The force exerted by the spring can be calculated from the mass of the block using Newton's second law:
F = ma
Where m is the mass of the block (2kg) and a is the acceleration of the block.
Since the plank is frictionless, the force acting on the block is the force exerted by the spring (F) and the weight of the block (mg).
F = ma = kx + mg
Now we can combine the equations to solve for the spring constant:
kx + mg = ma
Let's rearrange the equation to solve for k:
k = (ma - mg) / x
Substituting the given values:
k = (2a - 2g) / 1 (1)
Now let's consider the energy conservation principle. The total mechanical energy of the system is conserved when there are no external forces acting on it. Initially, the system has only potential energy stored in the spring, as the block is at rest.
The potential energy stored in the spring is given by PE_spring = (1/2)kx^2.
The kinetic energy of the block when it leaves the plank is given by KE_block = (1/2)mv^2, where v is the velocity of the block when it leaves the plank.
At this point, all the potential energy from the spring is converted into kinetic energy of the block:
PE_spring = KE_block
(1/2)kx^2 = (1/2)mv^2
Plugging in the value of k from equation (1):
(1/2)((2a - 2g)/1)(1^2) = (1/2)(2)v^2
Simplifying the equation:
(a - g) = v^2
This equation tells us that the difference between the acceleration of the block and the acceleration due to gravity is equal to the square of the velocity of the block when it leaves the plank.
Since the plank is frictionless, the acceleration of the plank is equal to the acceleration of the block.
Therefore, the velocity of the plank when the block leaves the plank is equal to the square root of the difference between the acceleration due to gravity and the acceleration of the block:
Velocity of the plank = sqrt(a - g)
To find the value of a (acceleration of the block), we can use Newton's second law:
ma = kx + mg
Substituting the value of k from equation (1):
2a = 2a - 2g + 2g
2a = 2a
This equation shows that the acceleration of the block is canceled out, meaning it doesn't affect the motion of the system.
Therefore, the velocity of the plank when the block leaves the plank is simply:
Velocity of the plank = sqrt(a - g) = sqrt(-g) = sqrt(9.8 m/s^2) ≈ 3.13 m/s
So, the velocity of the plank when the block leaves the plank is approximately 3.13 m/s.