A child’s toy consists of plastic attached to

a spring. The spring is compressed against
the floor a distance of 1.48 cm, and the toy is
released.
The acceleration of gravity is 9.8 m/s2 .
If the toy has a mass of 109 g and rises to
a maximum height of 58.7 cm, estimate the
force constant of the spring.
Answer in units of N/m

To estimate the force constant of the spring, we can use the principle of conservation of mechanical energy. The potential energy stored in the spring when compressed is converted into the potential energy when the toy reaches its maximum height.

First, let's convert the given quantities into SI units:
- Compressed distance: 1.48 cm = 0.0148 m
- Maximum height: 58.7 cm = 0.587 m
- Mass of the toy: 109 g = 0.109 kg

We know that the potential energy stored in the compressed spring is given by:
PE = (1/2)kx², where PE is the potential energy, k is the force constant of the spring, and x is the compression distance.

Similarly, the potential energy at the maximum height is given by:
PE = mgh, where m is the mass of the toy, g is the acceleration due to gravity, and h is the maximum height.

Setting these two expressions equal to each other, we can solve for the force constant of the spring (k):

(1/2)kx² = mgh

Rearranging the equation, we get:

k = (2mgh) / x²

Plugging in the given values, we have:

k = (2 * 0.109 kg * 9.8 m/s² * 0.587 m) / (0.0148 m)²

Calculating this expression, we get:

k ≈ 1,777.02 N/m

Therefore, the estimated force constant of the spring is approximately 1,777.02 N/m.