A 0.5-kg block slides along a horizontal frictionless surface at 2m/s. It is brought to rest by compressing a very long spring of spring constant 800N/m. The maximum spring compression is:
Well, well, well, looks like this little block had a wild ride! Now it's time to find out just how much that spring is going to squish.
To solve this problem, we can use the fact that the work done by the spring is equal to the work done by the block's initial kinetic energy.
So, the initial kinetic energy of the block is given by:
KE_initial = (1/2) * mass * velocity^2
Plug in the numbers:
KE_initial = (1/2) * 0.5 kg * (2 m/s)^2
KE_initial = 0.5 J
Now, the work done by the spring, which is equal to the maximum potential energy stored in the spring, is given by:
PE_spring = (1/2) * k * x^2
Where k is the spring constant and x is the maximum spring compression we're trying to find.
Since the block is brought to rest, all of its initial kinetic energy is transferred to the spring. So we have:
KE_initial = PE_spring
0.5 J = (1/2) * 800 N/m * x^2
Solving for x^2:
x^2 = (0.5 J) / (400 N/m)
x^2 = 0.00125 m
Taking the square root of both sides, we get:
x = 0.035 m
So, the maximum spring compression is 0.035 meters.
Just remember, this spring might feel a little squished, but trust me, it bounces back!
To determine the maximum spring compression, we need to use the principle of conservation of mechanical energy. This principle states that the initial mechanical energy of the system (block plus spring) is equal to the final mechanical energy.
The initial mechanical energy of the block is given by:
E_initial = (1/2) * m * v_initial^2
where
m = mass of the block (0.5 kg)
v_initial = initial velocity of the block (2 m/s)
Substituting the given values into the equation:
E_initial = (1/2) * 0.5 kg * (2 m/s)^2
= 0.5 J
Since the surface is frictionless, there is no work done by friction, and hence, no energy is lost due to friction.
The final mechanical energy of the system is given by:
E_final = (1/2) * k * x^2
where
k = spring constant (800 N/m)
x = maximum spring compression (unknown)
Substituting the given values into the equation:
E_final = (1/2) * 800 N/m * x^2
= 400x^2 J
Since mechanical energy is conserved:
E_initial = E_final
0.5 J = 400x^2 J
Simplifying the equation:
400x^2 = 0.5
Dividing both sides by 400:
x^2 = 0.5/400
Taking the square root of both sides:
x = sqrt(0.5/400)
x = 0.035 m
Therefore, the maximum spring compression is 0.035 meters.
To find the maximum spring compression, we can use the principle of conservation of mechanical energy. Initially, the block has kinetic energy while sliding, and when it comes to rest, that energy is completely converted to potential energy stored in the compressed spring.
The formula for potential energy stored in a spring is:
Potential Energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the displacement or compression of the spring.
Given that the block has a mass of 0.5 kg, the initial kinetic energy (KE) can be calculated using the formula:
Kinetic Energy (KE) = (1/2) * m * v^2
where m is the mass of the block and v is its velocity.
Substituting the known values:
KE = (1/2) * 0.5 kg * (2 m/s)^2
= (1/2) * 0.5 kg * 4 m^2/s^2
= 1 J
Since energy is conserved, the potential energy stored in the spring when the block comes to rest must be equal to 1 J. Thus, we have:
1 J = (1/2) * k * x^2
Rearranging the equation, we can solve for x:
2 J = k * x^2
x^2 = 2 J / k
x^2 = 2 J / (800 N/m)
x^2 = 0.0025 m^2/N
Taking the square root of both sides, we find:
x = 0.05 m
Therefore, the maximum spring compression is 0.05 meters, or 5 centimeters.
Fb = M*g = 0.5 * 9.8 = 4.9 N. = Force of block.
d = 4.9N/800N * 1m = 0.006125 m. = 0.6125 cm.