A child’s toy consists of plastic attached to
a spring. The spring is compressed against
the floor a distance of 2.41 cm, and the toy is
released.
The acceleration of gravity is 9.8 m/s
2
.
If the toy has a mass of 66 g and rises to
a maximum height of 57.3 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 energy. The potential energy stored in the spring when it is compressed is equal to the potential energy gained by the toy when it reaches its maximum height.
1. First, let's convert the given measurements to SI units:
- The distance the spring is compressed, 2.41 cm, can be converted to meters by dividing by 100: 2.41 cm = 0.0241 m.
- The maximum height reached by the toy, 57.3 cm, can be converted to meters using the same method: 57.3 cm = 0.573 m.
2. Next, let's calculate the potential energy of the compressed spring:
The potential energy stored in a spring is given by the equation: PE = (1/2) * k * x^2, where k is the force constant of the spring and x is the displacement from the equilibrium position.
Plugging in the values, we have: PE = (1/2) * k * (0.0241)^2.
3. Now, let's calculate the potential energy at the maximum height:
The potential energy at a certain height is given by the equation: PE = m * g * h, where m is the mass of the toy, g is the acceleration due to gravity, and h is the height.
Plugging in the values, we have: PE = 0.066 kg * 9.8 m/s^2 * 0.573 m.
4. Since the potential energy at the maximum height is equal to the potential energy of the compressed spring, we can set the two equations equal to each other and solve for the force constant, k:
(1/2) * k * (0.0241)^2 = 0.066 kg * 9.8 m/s^2 * 0.573 m.
Solve this equation for k to find the force constant of the spring.
Performing these calculations will give you the estimate for the force constant of the spring, in units of N/m.