A mass and spring are arranged on a horizontal, frictionless table. The spring constant is k = 450 N/m, and the mass is 5.4 kg. The block is pushed against the spring so that the spring is compressed an amount 0.35 m, and then it is released. Find the velocity of the mass when it leaves the spring.
potential energy stored in spring = kinetic energy of block
(1/2) k x^2 = (1/2) m v^2
450 (.35)^2 = 5.4 v^2
Well, this spring must have really bounced back! Let's find the velocity at which our brave mass leaves the spring.
Now, we know that the potential energy stored in a spring is given by the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, the spring constant is 450 N/m and the displacement is 0.35 m. Plugging these values into the formula, we get:
Potential Energy = (1/2) * 450 N/m * (0.35 m)^2
Simplifying this equation gives us:
Potential Energy = 27.56 J
Now, according to the law of conservation of energy, this potential energy will be converted entirely into kinetic energy as the mass leaves the spring:
Potential Energy = Kinetic Energy
So, the kinetic energy can be found using the formula:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the object passing through the spring and v is its velocity.
Plugging in the values, we get:
27.56 J = (1/2) * 5.4 kg * v^2
Simplifying further gives us:
27.56 J = 2.7 kg * v^2
Dividing both sides by 2.7 kg, we find:
v^2 = 10.207 J/kg
Taking the square root of both sides, we finally get:
v = 3.19 m/s (approximately)
So, the velocity at which our mass leaves the spring is approximately 3.19 m/s. Keep in mind that this calculation assumes no loss of energy due to other factors like air resistance or friction. Also, remember to only try this at home if you're a trained spring-whisperer!
To find the velocity of the mass when it leaves the spring, we can use the principle of conservation of mechanical energy. At the initial position, the spring is compressed and has potential energy stored in it, and at the final position, the mass has gained kinetic energy.
The potential energy stored in the spring is given by:
PE = 1/2 * k * x^2
where k is the spring constant and x is the compression of the spring.
The initial potential energy is given by:
PE_initial = 1/2 * 450 N/m * (0.35 m)^2
PE_initial = 1/2 * 450 N/m * 0.1225 m^2
PE_initial = 27.56 J
According to the conservation of mechanical energy, the total mechanical energy is conserved and equal to the sum of potential energy and kinetic energy:
Total mechanical energy = PE_initial + KE_final
Since the spring is released without any external forces acting on the system, the mechanical energy is conserved, and therefore, the kinetic energy at the final position is equal to the initial potential energy:
KE_final = 27.56 J
The kinetic energy is given by:
KE = 1/2 * m * v^2
where m is the mass of the object and v is its velocity.
Substituting the values, we have:
27.56 J = 1/2 * 5.4 kg * v^2
Solving for v, we get:
27.56 J = 1/2 * 5.4 kg * v^2
v^2 = (2 * 27.56 J) / 5.4 kg
v^2 = 10.18 J/kg
v ≈ √10.18 J/kg
v ≈ 3.19 m/s
Therefore, the velocity of the mass when it leaves the spring is approximately 3.19 m/s.
To find the velocity of the mass when it leaves the spring, we can use the principle of conservation of mechanical energy.
The initial potential energy stored in the compressed spring will be converted into kinetic energy of the mass when it is released. In this case, the potential energy of the spring is given by:
Potential energy (PE) = 1/2 * k * x^2
where k is the spring constant and x is the compression distance.
We can calculate the potential energy as follows:
PE = 1/2 * k * x^2 = 1/2 * 450 N/m * (0.35 m)^2
PE = 1/2 * 450 N/m * 0.1225 m^2
PE = 27.5625 J
Since the potential energy is converted into kinetic energy as the mass leaves the spring, we can equate the potential energy to the kinetic energy:
PE = KE
where KE is the kinetic energy.
From the equation for kinetic energy:
Kinetic energy (KE) = 1/2 * m * v^2
where m is the mass of the object and v is its velocity.
We can rearrange the equation to solve for velocity:
v = sqrt(2 * KE / m)
Plugging in the values:
v = sqrt(2 * 27.5625 J / 5.4 kg)
v = sqrt(51.25 m^2/s^2 / 5.4 kg)
v = sqrt(9.4907 m^2/s^2)
v ≈ 3.08 m/s
Therefore, the velocity of the mass when it leaves the spring is approximately 3.08 m/s.