A vertical spring (k = 1040 N/m) is compressed a distance of 11.6 cm. A 47.9 g mass is placed on the spring, then released so the mass flies upward. When the mass is 120 cm above the spring, what is the speed of the mass?
Let Y be the elevation from the compressed position. When released and after reaching elevation Y = 0.12 m, the kinetic energy is the initial spring potential energy minus the gain ib gravitational potential energy.
KE = (PE)spring - (PE)gravity
(1/2)(0.0479 kg)V^2 = (1/2)k(0.116m)^2 - (0.0479)(9.8 m/s^2)(0.12 m)
Solve for V
To find the speed of the mass when it is 120 cm above the spring, we can use the principle of conservation of energy.
The initial energy of the system is stored as potential energy in the compressed spring, and it is converted into kinetic energy as the mass moves upward.
Let's break down the problem into steps:
Step 1: Find the potential energy stored in the spring when it is compressed.
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 compression or extension of the spring.
In this case, k = 1040 N/m and x = 11.6 cm = 0.116 m.
So, the potential energy stored in the spring is given by:
Potential energy = (1/2) * 1040 * (0.116)^2
Step 2: Find the gravitational potential energy of the mass when it is 120 cm above the spring.
The gravitational potential energy of an object is given by the formula:
Gravitational potential energy = m * g * h
Where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
In this case, m = 47.9 g = 0.0479 kg, g = 9.8 m/s^2, and h = 120 cm = 1.2 m.
So, the gravitational potential energy of the mass is given by:
Gravitational potential energy = 0.0479 * 9.8 * 1.2
Step 3: Set up the conservation of energy equation.
According to the conservation of energy principle, the initial potential energy of the system is equal to the final kinetic energy of the mass.
Initial potential energy = Final kinetic energy
Step 4: Solve for the final kinetic energy using the conservation of energy equation.
Final kinetic energy = (1/2) * m * v^2
Where m is the mass of the object and v is the speed of the object.
Using the conservation of energy equation:
(1/2) * 1040 * (0.116)^2 = (1/2) * 0.0479 * v^2
Step 5: Solve for the speed, v.
Divide both sides of the equation by (1/2) * 0.0479:
1040 * (0.116)^2 = v^2
v = sqrt(1040 * (0.116)^2)
By evaluating this expression, we can find the speed of the mass when it is 120 cm above the spring.