A child’s toy consists of plastic attached to
a spring. The spring is compressed against
the floor a distance of 2.55 cm, and the toy is
released.
The acceleration of gravity is 9.8 m/s
2
.
If the toy has a mass of 116 g and rises to
a maximum height of 64.4 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 potential energy formula and Hooke's Law.
First, let's calculate the gravitational potential energy when the toy reaches its maximum height. The formula for gravitational potential energy is:
PE_gravity = m * g * h
Where:
PE_gravity = gravitational potential energy
m = mass of the toy (converted to kg)
g = acceleration due to gravity (given as 9.8 m/s^2)
h = maximum height reached by the toy (converted to m)
Converting the mass of the toy from grams to kilograms:
Mass (m) = 116 grams / 1000 = 0.116 kg
Converting the maximum height from centimeters to meters:
Height (h) = 64.4 cm / 100 = 0.644 m
Substituting the values into the formula:
PE_gravity = 0.116 kg * 9.8 m/s^2 * 0.644 m
Next, let's determine the potential energy stored in the spring. The formula for potential energy stored in a spring is:
PE_spring = (1/2) * k * x^2
Where:
PE_spring = potential energy stored in the spring
k = force constant of the spring (what we need to find)
x = compression or extension of the spring (in meters)
In this case, the spring is compressed a distance of 2.55 cm, which is 0.0255 meters.
Substituting the values into the formula:
PE_spring = (1/2) * k * (0.0255 m)^2
Since the toy is released, the potential energy stored in the spring is equal to the gravitational potential energy when the toy reaches its maximum height:
PE_gravity = PE_spring
Therefore, we can set the two equations equal to each other:
0.116 kg * 9.8 m/s^2 * 0.644 m = (1/2) * k * (0.0255 m)^2
Simplifying the equation, we can solve for k:
k = (0.116 kg * 9.8 m/s^2 * 0.644 m) / [(1/2) * (0.0255 m)^2]
Now, we can perform the calculations to find the force constant (k):
k = (0.116 kg * 9.8 m/s^2 * 0.644 m) / [(1/2) * (0.0255 m)^2]
k ≈ 96.1 N/m
Therefore, the estimated force constant of the spring is approximately 96.1 N/m.