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
He moved it the correct way
To estimate the force constant of the spring, you can use the equation involving the potential energy of the spring and gravitational potential energy.
The equation is:
1/2 K x^2 = mgh
Where:
K is the force constant of the spring (in N/m)
x is the distance the spring is compressed (in meters)
m is the mass of the toy (in kilograms)
g is the acceleration due to gravity (in m/s^2)
h is the maximum height the toy rises to (in meters)
Plugging in the given values:
x = 0.0246 m
m = 0.102 kg
g = 9.8 m/s^2
h = 0.601 m
1/2 K (0.0246)^2 = 0.102 (9.8) (0.601)
Now, solve for K:
K = (0.102 × 9.8 × 0.601)/(0.5 × 0.0246^2)
K = 190418.8777 N/m (approx)
Therefore, the estimated force constant of the spring is 190418.8777 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 compressed spring is equal to the potential energy gained when the toy reaches its maximum height.
Step 1: Calculate the potential energy stored in the compressed spring.
The potential energy stored in a spring is given by the formula:
Potential energy = 0.5 * k * x^2
Where k is the force constant of the spring and x is the compression or extension of the spring.
In this case, the compression of the spring is 2.46 cm, which is equivalent to 0.0246 m.
The mass of the toy is 102 g, which is equivalent to 0.102 kg.
The acceleration due to gravity is 9.8 m/s^2.
Potential energy stored in the spring = 0.5 * k * (0.0246)^2
Step 2: Calculate the potential energy gained when the toy reaches its maximum height.
The potential energy gained when the toy reaches its maximum height is equal to the gravitational potential energy.
Gravitational potential energy = mass * gravity * height
In this case, the mass of the toy is 102 g, which is equivalent to 0.102 kg.
The acceleration due to gravity is 9.8 m/s^2.
The maximum height reached by the toy is 60.1 cm, which is equivalent to 0.601 m.
Potential energy gained when the toy reaches its maximum height = 0.102 * 9.8 * 0.601
Step 3: Equate the potential energy stored in the spring to the potential energy gained.
0.5 * k * (0.0246)^2 = 0.102 * 9.8 * 0.601
Step 4: Solve for the force constant, k.
k = (0.102 * 9.8 * 0.601) / (0.5 * (0.0246)^2)
k = 190418.8777 N/m
Therefore, the estimated force constant of the spring is 190418.8777 N/m.
Yes, ok, however it depends on what max height means, I would have assumed that is max height of the center of gravity of the toy, and the toy cg was the zero point for the 60cm measuremetn, but who knows?
E-4 means 1/10000, so you moved the decimal the wrong way.....