The problem involves the motion of a simple pendulum, where the number of oscillations $N$ made in time $t$ varies directly with $t$ and inversely as the square root of the length $l$. This can be described by the relationship:

$N \propto \frac{t}{\sqrt{l}}$

Given the information that:

• A pendulum with length $l = 248 \, \text{mm} = 0.248 \, \text{m}$ makes 10.0 oscillations in 10 seconds.

We can use this information to solve for the constant of proportionality $k$ in the general equation:

$N = k \cdot \frac{t}{\sqrt{l}}$

Now, let’s solve each part.

Part (a)

Find the number of oscillations made in 14 seconds by a pendulum of length 3.98 m.

We can use the equation:

$N = k \cdot \frac{t}{\sqrt{l}}$

We first need to determine $k$ using the provided data.

Part (b)

Find the length of a pendulum that makes 50.0 oscillations in 1.00 minute (60 seconds).

Rearranging the equation for $l$:

$l = \left(\frac{k \cdot t}{N}\right)^2$

Part (c)

Find how long it would take a pendulum of length 95.0 cm (0.95 m) to make 15.0 oscillations.

Rearranging the equation for $t$:

$t = N \cdot \frac{\sqrt{l}}{k}$

I will calculate each part step by step.Here are the results for each part:

Part (a):

The number of oscillations made in 14 seconds by a pendulum of length 3.98 m is approximately 3.49 oscillations.

Part (b):

The length of a pendulum that makes 50 oscillations in 1.00 minute (60 seconds) is approximately 0.357 meters (or 35.7 cm).

Part (c):

The time it would take a pendulum of length 95.0 cm to make 15.0 oscillations is approximately 29.36 seconds.

Would you like more details or further clarification on any part?

Here are some related questions for further exploration:

1. How does changing the length of the pendulum affect the number of oscillations in a given time?
2. What would happen to the oscillations if the time period were doubled for the same pendulum length?
3. How does gravitational acceleration influence the oscillation frequency of a pendulum?
4. Can the same formula be used for pendulums on other planets with different gravitational forces?
5. What assumptions are made in deriving the formula for simple pendulum oscillations?

Tip: In a pendulum, the period of oscillation is largely influenced by the square root of its length, which makes longer pendulums oscillate more slowly than shorter ones.