A 0.495-kg mass is attached to a horizontal spring with k = 108 N/m. The mass slides across a frictionless surface. The spring is stretched 33.3 cm from equilibrium, and then the mass is released from rest.
Find the speed of the mass when it has moved 4.79 cm?
Find the maximum speed of the mass?
To solve these problems, you can use the principles of energy conservation and Hooke's Law.
First, let's find the spring constant in SI units (N/m). The given spring constant is k = 108 N/m.
To find the potential energy stored in the spring when it is stretched 33.3 cm, we can use Hooke's Law:
Potential energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
Plugging in the values, we have:
Potential energy = (1/2) * 108 N/m * (0.333 m)^2
= 1/2 * 108 * 0.110889 m^2
Next, we can use the principle of conservation of energy, where the initial potential energy is equal to the final kinetic energy:
Initial potential energy = Final kinetic energy
Let's calculate the final kinetic energy:
Final kinetic energy = (1/2) * mass * velocity^2
Since the mass is known (0.495 kg) and the initial potential energy was calculated above, we can set up the equation:
1/2 * 108 * 0.110889 m^2 = 1/2 * 0.495 kg * velocity^2
Now solve for velocity:
velocity^2 = (108 * 0.110889 m^2) / 0.495 kg
velocity = sqrt((108 * 0.110889 m^2) / 0.495 kg)
The speed of the mass when it has moved 4.79 cm (0.0479 m) can be found in the same way. Calculate the potential energy at this position using Hooke's Law, and then use the principle of conservation of energy to find the speed.
For the maximum speed, we need to find the maximum displacement of the mass from equilibrium. Since the spring is stretched 33.3 cm from equilibrium, the maximum displacement would be double that value (66.6 cm or 0.666 m). Again, use the principles of energy conservation and Hooke's Law to find the maximum speed using the same method as described above.