A small ball is attached to one end of a spring that has an unstrained length of 0.202 m. The spring is held by the other end, and the ball is whirled around in a horizontal circle at a speed of 3.18 m/s. The spring remains nearly parallel to the ground during the motion and is observed to stretch by 0.009 m. By how much would the spring stretch if it were attached to the ceiling and the ball allowed to hang straight down, motionless?

Answer is in m

To find the amount by which the spring would stretch if it were attached to the ceiling and the ball allowed to hang motionless, we can apply Hooke's Law.

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, it can be expressed as:

F = -k * x

Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

In this case, since the ball is whirled around in a horizontal circle and stretches the spring by 0.009 m, we can determine the spring constant using the formula:

F = k * x

Solving for k, we have:

k = F / x

The force exerted by the spring can be calculated using the centripetal force formula:

F = (m * v^2) / r

Where m is the mass of the ball, v is the speed of the ball, and r is the radius of the circle.

Given that the speed of the ball is 3.18 m/s, the radius of the circle is equal to the stretched length of the spring, which is 0.202 m.

Plugging in these values, we can calculate the force exerted by the spring:

F = (m * v^2) / r
F = (m * (3.18)^2) / 0.202

Now, we can substitute this value of F into the previous equation to find the spring constant:

k = F / x
k = [(m * (3.18)^2) / 0.202] / 0.009

Finally, to find the amount by which the spring would stretch if it were attached to the ceiling and the ball allowed to hang motionless, we can determine the displacement x' from the equilibrium position when the ball is motionless:

x' = F / k
x' = 0 / k

Since the ball is motionless, the force exerted by the spring is zero. Therefore, the displacement x' will also be zero.

So, the spring would not stretch at all when attached to the ceiling and the ball is allowed to hang motionless. The answer is 0 m.

To solve this question, we need to understand the concept of centripetal force and the equilibrium of forces acting on the ball. Let's break down the steps to find the answer:

Step 1: Find the applied centripetal force to keep the ball moving in a horizontal circle.
The centripetal force required to keep an object moving in a circle can be calculated using the formula:
Centripetal Force = (mass * velocity^2) / radius

Given that the speed of the ball is 3.18 m/s and the radius is equal to the unstrained length of the spring, which is 0.202 m, we can calculate the centripetal force.

Centripetal Force = (mass * (3.18 m/s)^2) / 0.202 m

Step 2: Find the increase in the weight of the ball due to the centripetal force.
When the ball is whirled around in the horizontal circle, the spring stretches due to the increase in the weight of the ball caused by the centripetal force. The increase in weight can be calculated using Hooke's Law:
F = kx, where F is the force exerted on the spring, k is the spring constant, and x is the displacement (stretch) of the spring.

Given that the stretch of the spring is 0.009 m, we can write the equation as:
Centripetal Force = k * 0.009 m

Step 3: Set up an equation using the weight of the ball and the centripetal force.
The weight of the ball is given by the equation:
Weight = mass * gravity

In equilibrium, the weight of the ball should balance the centripetal force. Therefore, we can equate the weight of the ball to the centripetal force:
Weight = Centripetal Force

Step 4: Calculate the stretch of the spring when the ball is motionless hanging from the ceiling.
When the ball is motionless, the weight of the ball will be equal to the centripetal force. Using the equation from step 3 and rearranging, we can solve for the stretch of the spring when the ball is hanging motionless.

Stretch of the spring = Weight / k

Step 5: Calculate the answer using the given values.
Substitute the known values into the equation from step 4 to find the stretch of the spring when the ball is hanging motionless:
Stretch of the spring = (mass * gravity) / k

The given values needed to solve the equation are:
Mass = unknown (not given)
Gravity = 9.8 m/s^2
k = unknown (not given)

Since the mass and the spring constant are not provided, we cannot calculate the exact stretch of the spring when the ball is hanging motionless.

i think that we should do the sum of length to obtain the result.