A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end is fixed to a wall. 3.0 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 16 m/s2.
(a) Find the value of the spring constant.
(b) Find the value of the mass.
(a) Sure, I would be happy to find the value of the spring constant for you. But first, let me ask you a question: Why did the scarecrow win an award? Because he was outstanding in his field! Now, let's get back to the problem.
The work done on the spring is given by the formula:
Work = (1/2) k x²
Where k is the spring constant and x is the displacement.
Given that the work done is 3.0 J and the displacement is 0.17 m, we can plug these values into the formula and solve for k:
3.0 J = (1/2) k (0.17 m)²
Now, let me calculate this for you... *calculating noises* ...and the value of the spring constant is 66.18 N/m!
(b) Now, let's move on to finding the value of the mass. To do this, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement:
F = -kx
Where F is the force, k is the spring constant, and x is the displacement.
At maximum acceleration, the net force on the mass is given by:
F_net = m * a_max
Where m is the mass and a_max is the maximum acceleration.
We can equate these two equations:
-m * a_max = -k * x
Since we have already found the values for k and x, we can plug these in and solve for the mass:
m * 16 m/s² = 66.18 N/m * 0.17 m
And after performing the calculations... *calculating noises* ...the mass equals 1.98 kg.
So, the spring constant is 66.18 N/m and the mass is 1.98 kg. I hope my answers made you spring into action with delight!
(a) To find the value of the spring constant, we can use the formula for the potential energy stored in a spring:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement of the spring.
Given that 3.0 J of work is required to compress the spring by 0.17 m, we can equate the work done to the potential energy stored in the spring:
3.0 J = (1/2)k(0.17 m)^2
Simplifying the equation:
3.0 J = (1/2)k(0.0289 m^2)
Multiplying both sides by 2 and dividing both sides by 0.0289 m^2:
k = (3.0 J) / (2 * 0.0289 m^2)
k = 52.164 N/m
Therefore, the value of the spring constant is 52.164 N/m.
(b) To find the value of the mass, we can use Newton's second law of motion:
F = ma
The force exerted by the spring is given by Hooke's law:
F = -kx
where F is the force, k is the spring constant, and x is the displacement.
At the maximum acceleration, the net force acting on the mass is equal to the force exerted by the spring:
F = ma = -kx
Rearranging the equation:
m = -kx / a
Substituting the values we know:
m = - (52.164 N/m)(0.17 m) / 16 m/s^2
m = - 0.88548 kg
Since mass cannot have a negative value, the value of the mass is approximately 0.885 kg.
To find the spring constant and the mass, we can use the formulas for elastic potential energy and force.
(a) The elastic potential energy stored in a spring is given by the formula:
Elastic Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the mass from its equilibrium position. We are given that the work done to compress the spring is 3.0 J and the displacement is 0.17 m. Substituting these values into the formula, we get:
3.0 J = (1/2) * k * (0.17 m)^2
Simplifying the equation, we have:
3.0 J = (1/2) * k * 0.0289 m^2
6 J = k * 0.0289 m^2
Now, we can solve for k:
k = 6 J / 0.0289 m^2
k ≈ 207.612 J/m
Therefore, the value of the spring constant is approximately 207.612 J/m.
(b) The acceleration of an object connected to a spring is related to the force exerted by the spring through Newton's second law:
F = m * a
where F is the force, m is the mass, and a is the acceleration. In this case, the maximum acceleration is given as 16 m/s^2. The force exerted by the spring can be calculated using Hooke's Law:
F = k * x
where F is the force, k is the spring constant, and x is the displacement. The maximum force occurs when the spring is at maximum compression, which is 0.17 m. So:
F = k * 0.17 m
Setting the two equations for force equal to each other, we have:
m * a = k * 0.17 m
Now we can solve for the mass:
m = (k * 0.17 m) / a
m ≈ (207.612 J/m * 0.17 m) / 16 m/s^2
m ≈ 2.201 kg
Therefore, the value of the mass is approximately 2.201 kg.