A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. 8.0 J of work is required to compress the spring by 0.17 m. If the mass is released from rest with the spring compressed, it experiences a maximum acceleration of 11 m/s2.
(a) Find the value of the spring constant.
(b) Find the value of the mass.
To find the value of the spring constant and the value of the mass, we can use the concepts of work and energy, as well as the laws of motion.
(a) Finding the spring constant (k):
We know that the work done on the spring (W) is equal to the potential energy stored in the spring (PE):
W = PE
The formula for the potential energy stored in a spring is given by:
PE = (1/2) * k * x^2
Where k is the spring constant and x is the displacement of the spring from its equilibrium position.
Given:
W = 8.0 J (work done on the spring)
x = 0.17 m (displacement of the spring)
We can substitute the given values into the equation to find the spring constant:
8.0 J = (1/2) * k * (0.17 m)^2
16 J = k * 0.0289 m^2
k = 16 J / (0.0289 m^2)
k = 554.67 N/m
Therefore, the value of the spring constant is approximately 554.67 N/m.
(b) Finding the mass (m):
We can use Newton's second law of motion to find the mass of the object attached to the spring. The maximum acceleration (a) experienced by the mass when released is related to the net force acting on it (F_net) and the mass (m) according to the equation:
F_net = m * a
The net force acting on the mass is the force exerted by the spring, which is given by Hooke's law:
F_spring = -k * x
Where F_spring is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.
Let's calculate the net force acting on the mass:
F_spring = -k * x
F_spring = -554.67 N/m * 0.17 m
F_spring = -94.2 N (Note: The negative sign indicates that the force by the spring is in the opposite direction to the compression)
The net force acting on the mass is equal to the mass (m) multiplied by the maximum acceleration (a):
F_net = m * a
-94.2 N = m * 11 m/s^2
m = -94.2 N / 11 m/s^2
m = -8.55 kg
Since mass cannot be negative, we can ignore the negative sign and take the magnitude of the mass.
Therefore, the value of the mass is approximately 8.55 kg.
when compressed work=1/2 k x^2 solve for k.
Force=kx=ma solve for mass m