9. A block attached to a spring of negligible mass undergoes simple harmonic motion on a frictionless horizontal surface. The potential energy of the system is zero at the equilibrium position and has a maximum value of 50 J. When the displacement of the block is half the minimum value its instantaneous kinetic energy is

a.) 0 J
b.) 12.5 J
c.) 25 J
d.) 37.5 J
e.) 50 J

37.5

Well, well, well, looks like we're diving into some physics! Brace yourself for some clowny wisdom.

Alright, let's break this down. We're talking about a block attached to a spring, going back and forth, just like a spring chicken. The potential energy of the system is zero at equilibrium position, so the block feels no potential pressure, just like a retired acrobat chilling on a hammock.

Now, when the displacement of the block is half the minimum value, we can assume the block is halfway between its maximum compression and maximum expansion. That means it's not too squished or too stretched, just like a pair of clown pants that fit just right!

Since the potential energy is at its maximum when the kinetic energy is at its minimum, and both of them sum up to a grand total of 50 J, it means the kinetic energy at that halfway point is also 50 J!

So, the answer is e.) 50 J. The block is rockin' and rollin' with an energetic dance routine.

To determine the answer, we need to make use of the conservation of mechanical energy in simple harmonic motion. The total mechanical energy, which is the sum of the potential energy and kinetic energy, remains constant throughout the motion.

Given that the potential energy has a maximum value of 50 J, we can conclude that the total mechanical energy is also 50 J at any point during the motion.

When the displacement of the block is half the minimum value, it means that it has traveled half of the amplitude. At this position, the potential energy is zero (since it is at the equilibrium position), which means that all the mechanical energy is converted to kinetic energy.

Therefore, the instantaneous kinetic energy is equal to the total mechanical energy, which is 50 J.

Hence, the answer is e.) 50 J.

To find the answer, let's break down the problem step by step:

Step 1: Understand the situation
We have a block attached to a spring, which undergoes simple harmonic motion on a frictionless horizontal surface. The potential energy of the system is zero at the equilibrium position and has a maximum value of 50 J.

Step 2: Analyze the displacement of the block
When the displacement of the block is half the minimum value, it means the block has moved from the equilibrium position to a point halfway towards the maximum displacement. This can be represented as x = -A/2, where A is the maximum displacement of the block.

Step 3: Analyze the potential energy
Given that the potential energy of the system is zero at the equilibrium position and has a maximum value of 50 J, we can calculate the potential energy at this specific displacement using the equation for potential energy in a spring-mass system:

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

Where k is the spring constant and x is the displacement from the equilibrium position.

Since the potential energy is zero at the equilibrium position, we only need to consider the potential energy when the block is at the specific displacement x = -A/2:

PE = (1/2) * k * (-A/2)^2
PE = (1/2) * k * (A^2/4)
PE = (k * A^2) / 8

Given that the maximum potential energy is 50 J, we can set up the equation:

50 J = (k * A^2) / 8

Step 4: Find the kinetic energy
In simple harmonic motion, the total mechanical energy (E) of the system remains constant. It is the sum of kinetic energy (KE) and potential energy (PE):

E = KE + PE

Since the potential energy is at its maximum value of 50 J when the displacement is half the minimum value, we can assume that the total mechanical energy is also 50 J:

E = KE + 50 J

Now, let's solve for the kinetic energy when the displacement is x = -A/2:

E = KE + PE
50 J = KE + (k * A^2) / 8

To find the kinetic energy (KE), we rearrange the equation:

KE = 50 J - (k * A^2) / 8

Step 5: Substitute for k and A
Unfortunately, we don't have the values for k and A, so we cannot determine the exact numerical value of the kinetic energy. However, we can still determine its relationship to the potential energy.

Looking at the equation for KE, we can see that it depends on the square of the maximum displacement (A) and the spring constant (k). As the maximum displacement (A) increases, the kinetic energy will also increase proportionally. But we cannot determine the specific value without knowing the values of k and A.

Therefore, the answer is not any of the given options (a.) 0 J, b.) 12.5 J, c.) 25 J, d.) 37.5 J, e.) 50 J).