A spring (k = 400 N/m) is hung vertically.
a) If a 5 kg mass is attached to the end of the spring and gently lowered to rest position, what will be the stretch of the spring?
b) If the 5 kg mass is attached and simply dropped, what will be the maximum velocity and the maximum stretch?
c) What will be the period of resulting oscillation?
To find the answers, we'll need to use the formulas and principles of spring motion:
a) To find the stretch of the spring when a 5 kg mass is gently lowered to the rest position, we can use Hooke's Law: F = -kx, where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the force applied to the spring due to gravity is equal to the weight of the mass: F = mg, where m is the mass and g is the acceleration due to gravity. The negative sign in Hooke's Law indicates that the force applied by the spring is in the opposite direction of the displacement.
So, setting the force due to gravity equal to the force applied by the spring, we have:
mg = kx
Plugging in the values:
m = 5 kg
g = 9.8 m/s^2 (approximate value)
k = 400 N/m
Rearranging the equation to solve for x, we get:
x = (mg) / k
Substituting the values:
x = (5 kg * 9.8 m/s^2) / 400 N/m
Now, calculate x to find the stretch of the spring.
b) When the 5 kg mass is simply dropped, it will undergo an oscillatory motion called simple harmonic motion. At the maximum displacement or stretch, the mass's velocity will be zero.
To find the maximum velocity of the mass, we can use energy conservation. When the mass is at its highest point, all of its potential energy will be converted into kinetic energy. The potential energy of a spring is given by: PE = (1/2)kx^2.
When the mass is at its maximum displacement, its potential energy is at a maximum, while its kinetic energy is zero:
PE = (1/2)kx^2 = mgh
Solving for x, we have:
x = sqrt((2mgh) / k)
Plugging in the values:
m = 5 kg
g = 9.8 m/s^2 (approximate value)
k = 400 N/m
h = maximum stretch (which we will calculate in the next step)
Now, calculate x to find the maximum stretch.
c) The period of oscillation can be calculated using the formula:
T = 2π * sqrt(m / k)
Substituting the values:
m = 5 kg
k = 400 N/m
Now, calculate T to find the period of oscillation.
By following these steps, you should be able to find the answers to these questions.