A child’s toy consists of plastic attached to a spring. The spring is compressed against the floor a distance of 2.46 cm, and the toy is released. The acceleration of gravity is 9.8 m/s2 . If the toy has a mass of 102 g and rises to a maximum height of 60.1 cm, estimate the force constant of the spring. Answer in units of N/m.
My work:
Spring potential = gravitational potential
.5 (K)(.0246)^2=.102 (9.8)(57.64)
K=57.616944/3.0258e-4
K=190418.8777
To estimate the force constant of the spring, let's first understand the problem and the given information.
The toy is released from a compressed position and rises to a maximum height. We can assume that all the potential energy in the spring is converted into gravitational potential energy as the toy rises.
Given information:
- Distance the spring is compressed, d = 2.46 cm = 0.0246 m
- Mass of the toy, m = 102 g = 0.102 kg
- Acceleration due to gravity, g = 9.8 m/s^2
- Maximum height reached, h = 60.1 cm = 0.601 m
Now, let's set up the equation using the conservation of energy:
Potential energy stored in the spring = Gravitational potential energy at maximum height
0.5 * k * d^2 = m * g * h
Where:
- k is the force constant of the spring (in N/m)
- d is the distance the spring is compressed (in m)
- m is the mass of the toy (in kg)
- g is the acceleration due to gravity (in m/s^2)
- h is the maximum height reached by the toy (in m)
Now, substitute the given values:
0.5 * k * (0.0246)^2 = 0.102 * 9.8 * 0.601
Simplifying the equation:
0.5 * k * 0.000602 = 0.600996
Now, solve for k:
k = 0.600996 / (0.5 * 0.000602)
k = 190418.8777 N/m
Therefore, the estimate for the force constant of the spring is 190418.8777 N/m.