An object with a mass of 5.13 kg is placed on top of a spring and compresses the spring by 0.25 m. (a) What is the force constant of the spring? (b) How high will this object go when the spring releases its energy?
(a) The force constant of the spring can be calculated using the equation F = kx, where F is the force, k is the force constant, and x is the displacement. In this case, F = 5.13 kg x 9.8 m/s2 = 50.6 N and x = 0.25 m. Therefore, the force constant of the spring is k = 50.6 N/0.25 m = 202.4 N/m.
(b) The height to which the object will rise when the spring releases its energy can be calculated using the equation h = (1/2)kx2, where h is the height, k is the force constant, and x is the displacement. In this case, h = (1/2) x 202.4 N/m x (0.25 m)2 = 1.26 m. Therefore, the object will rise to a height of 1.26 m when the spring releases its energy.
To find the force constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position. Hooke's Law can be written as:
F = -kx
where F is the force, k is the force constant, and x is the displacement.
(a) To find the force constant of the spring, we need to find the force exerted by the spring. Since the object is placed on top of the spring and compresses it, the force exerted by the spring is equal to the weight of the object. The weight can be calculated by multiplying the mass (m) of the object by the acceleration due to gravity (g). The equation for weight is:
Weight = mg
Given that the mass of the object is 5.13 kg, and assuming that the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight as:
Weight = 5.13 kg * 9.8 m/s^2 = 50.274 N
Now we can substitute the force and displacement into Hooke's Law equation to find the force constant:
50.274 N = -k * 0.25 m
Rearranging the equation to solve for k:
k = -50.274 N / 0.25 m
k ≈ -201.096 N/m (rounded to three decimal places)
The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.
Therefore, the force constant of the spring is approximately 201.096 N/m.
(b) To determine how high the object will go when the spring releases its energy, we need to consider the conservation of energy. When the spring is released, the potential energy stored in the spring is converted into gravitational potential energy as the object rises.
The potential energy stored in the spring can be calculated using the formula:
Potential energy = (1/2) * k * x^2
where k is the force constant and x is the displacement.
Since the object was compressed by 0.25 m, we can substitute the values into the formula:
Potential energy = (1/2) * 201.096 N/m * (0.25 m)^2
Evaluating the expression:
Potential energy = (1/2) * 201.096 N/m * 0.0625 m^2
Potential energy ≈ 6.329 N*m (rounded to three decimal places)
Now, we equate this potential energy to the gravitational potential energy of the object when it reaches its maximum height:
6.329 N*m = m * g * h
Solving for h (the maximum height), we divide both sides of the equation by mg:
h = 6.329 N*m / (5.13 kg * 9.8 m/s^2)
h ≈ 0.124 m (rounded to three decimal places)
Therefore, the object will rise to approximately 0.124 m (or 12.4 cm) when the spring releases its energy.
(a) To find the force constant of the spring, we can use Hooke's law, which states that the force applied to a spring is directly proportional to the displacement of the spring from its equilibrium position.
Hooke's law can be written as F = -kx, where F is the force applied to the spring, k is the force constant of the spring, and x is the displacement from the equilibrium position.
Given that the spring is compressed by 0.25 m, and using the equation F = -kx, we can rearrange the equation to solve for the force constant:
k = -F / x
Since the object is at rest on top of the spring, the force applied to the spring is equal to the gravitational force acting on the object, which can be calculated using the formula F = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we have:
k = - (5.13 kg * 9.8 m/s^2) / 0.25 m
Simplifying the equation:
k = - 50.274 N/m
Since the force constant of a spring is always positive, we can take the absolute value of k:
k ≈ 50.274 N/m
Therefore, the force constant of the spring is approximately 50.274 N/m.
(b) To determine the height the object will reach when the spring releases its energy, we can use the principle of conservation of mechanical energy.
The potential energy stored in the compressed spring is equal to the potential energy gained by the object as it moves upward against gravity.
The potential energy stored in a spring can be calculated using the equation U = (1/2)kx^2, where U is the potential energy, k is the force constant of the spring, and x is the displacement from the equilibrium position.
Since the object compresses the spring by 0.25 m, and using the equation U = (1/2)kx^2, we can calculate the potential energy stored in the spring:
U = (1/2)(50.274 N/m)(0.25 m)^2
Simplifying the equation:
U ≈ 0.78 J
This potential energy will be converted into gravitational potential energy as the object moves upward.
The gravitational potential energy can be calculated using the equation U = mgh, where U is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height.
Setting the potential energies equal to each other:
mgh = 0.78 J
Solving for h:
h = 0.78 J / (5.13 kg * 9.8 m/s^2)
Simplifying the equation:
h ≈ 0.016 m
Therefore, the object will rise to a height of approximately 0.016 m when the spring releases its energy.