In an arcade game a 0.087 kg disk is shot
across a frictionless horizontal surface by com-
pressing it against a spring and releasing it.
If the spring has a spring constant of
163 N/m and is compressed from its equi-
librium position by 4 cm, find the speed with
which the disk slides across the surface.
Answer in units of m/s.
uggghhhh i need help on this one too
To find the speed with which the disk slides across the surface, we can use the principle of conservation of mechanical energy. The initial mechanical energy stored in the compressed spring gets converted into the kinetic energy of the moving disk.
1. Calculate 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 displacement of the spring from its equilibrium position.
In this case, k = 163 N/m and x = 0.04 m
Potential Energy = (1/2) * 163 N/m * (0.04 m)^2
2. Now, equate the potential energy stored in the compressed spring to the kinetic energy of the moving disk:
Potential Energy = Kinetic Energy
(1/2) * 163 N/m * (0.04 m)^2 = (1/2) * m * v^2
Here, m is the mass of the disk and v is the speed of the disk.
The mass of the disk is given as 0.087 kg.
3. Rearrange the equation to solve for v:
v^2 = (163 N/m * (0.04 m)^2) / 0.087 kg
Take the square root of both sides to get the value of v:
v = sqrt((163 N/m * (0.04 m)^2) / 0.087 kg)
4. Substitute the values and calculate v:
v = sqrt((163 * 0.04^2) / 0.087)
v ≈ 0.53 m/s
Therefore, the speed with which the disk slides across the surface is approximately 0.53 m/s.
To find the speed with which the disk slides across the surface, we can use the principle of conservation of mechanical energy. The initial potential energy stored in the spring is converted into kinetic energy as the disk slides across the surface.
The potential energy stored in the compressed spring can be calculated using the formula:
Potential energy (U) = (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, k = 163 N/m and x = 0.04 m (converted from cm to m).
U = (1/2) * 163 N/m * (0.04 m)^2
U = 0.1312 J
As the disk slides across the surface, this potential energy is converted into kinetic energy. The kinetic energy can be calculated using the formula:
Kinetic energy (K) = (1/2) * m * v^2
where m is the mass of the disk and v is its velocity.
Setting the potential energy equal to the kinetic energy, we have:
0.1312 J = (1/2) * 0.087 kg * v^2
Simplifying,
0.2624 J = 0.087 kg * v^2
Dividing both sides of the equation by 0.087 kg,
v^2 = 3.0149 m^2/s^2
Taking the square root of both sides,
v = √(3.0149) m/s
v ≈ 1.7362 m/s
Therefore, the speed with which the disk slides across the surface is approximately 1.7362 m/s.