A child's toy consists of a piece of plastic attached to a spring. The spring is compressed against the floor a distance of 2.00 cm, and the toy is released. If the toy has a mass of 105 g and rises to a maximum height of 75.0 cm, estimate the force constant of the spring.
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To estimate the force constant of the spring, we can make use of Hooke's Law. Hooke's Law states that the force required to compress or extend a spring is directly proportional to the displacement of the spring from its equilibrium position.
The formula for Hooke's Law is given as:
F = -kx
Where:
F is the force applied on the spring,
k is the force constant or spring constant,
x is the displacement of the spring from its equilibrium position.
In this case, the spring is compressed by 2.00 cm or 0.02 m and released, causing the toy to rise to a maximum height of 75.0 cm or 0.75 m.
From the given information, we can determine:
x = 0.02 m (displacement)
F = m * g (force applied)
The mass of the toy is 105 g or 0.105 kg, and the acceleration due to gravity is approximately 9.8 m/s².
So, we have:
F = 0.105 kg * 9.8 m/s² = 1.029 N
Rearranging the formula for Hooke's Law, we get:
k = -F / x
Plugging in the values, we have:
k = -1.029 N / 0.02 m
Calculating this, we find:
k ≈ -51.45 N/m
As force constant (spring constant) cannot be negative, we take the magnitude, and the approximate force constant of the spring is 51.45 N/m.