A(n) 96.1 g ball is dropped from a height of 59.1 cm above a spring of negligible mass.The ball compresses the spring to a maximum displacement of 4.75403 cm.

The acceleration of gravity is 9.8 m/s^2.

Calculate the spring force constant k.
Answer in units of N/

Well, in order to find the spring force constant k, we can use Hooke's Law which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

So, F = -kx

Where:
F is the force exerted by the spring,
k is the spring force constant, and
x is the displacement of the spring.

Now, we can use the given information to calculate k.

First, let's convert the given values into meters:
Height of drop = 59.1 cm = 0.591 m
Maximum displacement = 4.75403 cm = 0.0475403 m

Next, let's find the gravitational potential energy of the ball when it is dropped:
PE = mgh
= (96.1 g)(9.8 m/s^2)(0.591 m)
= 344.86098 J

In this case, the gravitational potential energy is converted into the elastic potential energy of the spring when it is compressed. Therefore, we can equate the two:
PE = 0.5kx^2

Substituting the values:
344.86098 J = 0.5k(0.0475403 m)^2

Now we can solve for k:
k = (2 * 344.86098 J) / (0.0475403 m^2)
= 1449810.4376 N/m

So, the spring force constant is approximately 1,449,810.4376 N/m. (Although I don't know how springs feel about numbers that long... They might start bouncing back!)

To calculate the spring force constant (k), we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

First, let's convert the height and maximum displacement to meters:
Height (h) = 59.1 cm = 0.591 m
Maximum displacement (x) = 4.75403 cm = 0.0475403 m

Next, we need to calculate the potential energy of the ball at the maximum displacement. The potential energy is given by the equation:

Potential energy (PE) = m * g * h

where:
m = mass of the ball
g = acceleration due to gravity
h = height

PE = 96.1 g * 9.8 m/s^2 * 0.591 m

Now, since the potential energy is equal to the spring potential energy when the ball reaches maximum compression, we can equate it to the equation for spring potential energy:

PE = (1/2) * k * x^2

where:
k = spring force constant
x = maximum displacement of the spring

(1/2) * k * x^2 = 96.1 g * 9.8 m/s^2 * 0.591 m

Now, we can solve for k. Dividing both sides of the equation by (1/2) * x^2:

k = (96.1 g * 9.8 m/s^2 * 0.591 m) / (0.5 * x^2)

Substituting the given values:

k = (96.1 * 0.001 kg * 9.8 m/s^2 * 0.591 m) / (0.5 * (0.0475403 m)^2)

Simplifying the expression:

k = (0.056167 kilogram*meter/second^2)/m^2

Finally, expressing the answer in the desired unit:

k = 0.056167 N/m

Therefore, the spring force constant k is 0.056167 N/m.

To calculate the spring force constant, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is:

F = -kx

Where:
F is the force exerted by the spring,
k is the spring force constant, and
x is the displacement of the spring.

In this case, we need to find the spring force constant (k). The displacement (x) is given as 4.75403 cm = 0.0475403 m. The force (F) can be calculated using the weight of the ball.

The weight of the ball is given by the formula:

F = mg

Where:
m is the mass of the ball, and
g is the acceleration due to gravity.

In this case, the mass (m) is given as 96.1 g = 0.0961 kg, and the acceleration due to gravity (g) is 9.8 m/s^2.

Substituting the values into the formula, we get:

F = (0.0961 kg)(9.8 m/s^2)

Simplifying, we find:

F = 0.94018 N

Now, we can substitute the values of the force (F) and displacement (x) into Hooke's Law to find the spring force constant (k):

0.94018 N = -k(0.0475403 m)

Rearranging the equation to solve for k, we get:

k = -0.94018 N / 0.0475403 m

Calculating the value, we find:

k ≈ -19.7676 N/m

However, note that the spring force constant (k) is always positive. So, we take the absolute value to get the answer:

k ≈ 19.7676 N/m

Therefore, the spring force constant is approximately 19.7676 N/m.

total vertical fall of ball = .591 + .048 or

.639 meters
loss of potential energy by ball = .0961 * 9.81 * .639 = .602 Joules

so
(1/2) k (.04754)^2 = .602