A stone tied to the end of a string, 80 cm long, is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 seconds what is the magnitude and direction of the acceleration of the stone?

80cm=0.8m

T=25/14=1.786...seconds
v=(2pi*r)/T=2pi(0.8)/1.786...=2.815...m/s
a=v^2/r=2.815...^2/0.8=9.904m/s^2, towards the centre along the radius vector of the circle

9.9

To find the magnitude and direction of the acceleration of the stone, we need to use the following equations:

1. Centripetal acceleration: a = (v^2) / r
2. Tangential velocity: v = 2πr / T

Given:
- Length of the string (radius), r = 80 cm = 0.8 m
- Number of revolutions, n = 14
- Time taken for the revolutions, T = 25 seconds

Let's calculate the magnitude of the acceleration first:

1. Calculate the circumference of the circular path:
C = 2πr = 2π(0.8 m) ≈ 5.03 m

2. Calculate the total distance traveled by the stone:
Total distance, D = nC = 14(5.03 m) ≈ 70.42 m

3. Calculate the tangential velocity:
v = D / T = (70.42 m) / (25 s) ≈ 2.82 m/s

4. Calculate the centripetal acceleration:
a = (v^2) / r = (2.82 m/s)^2 / (0.8 m) ≈ 9.92 m/s^2

The magnitude of the acceleration is approximately 9.92 m/s^2.

Now, let's determine the direction of the acceleration:

The direction of the acceleration is always towards the center of the circle, which is the same as the direction of the net force. In this case, the centripetal acceleration is directed towards the center of the circle.

Therefore, the direction of the acceleration is towards the center of the circle.

To find the magnitude and direction of the acceleration of the stone, we need to calculate the centripetal acceleration.

The centripetal acceleration, denoted as \(a_c\), is given by the formula:

\[a_c = \frac{{v^2}}{{r}}\]

Where:
- \(v\) is the linear velocity of the stone
- \(r\) is the radius of the circular motion

First, we need to find the linear velocity \(v\) of the stone. Since we know the stone makes 14 revolutions in 25 seconds, we can calculate the linear velocity using the formula:

\[v = \frac{{2 \pi r}}{{T}}\]

Where:
- \(T\) is the time taken to complete one revolution

The time taken to complete one revolution, \(T\), is given by:

\[T = \frac{{\text{{total time}}}}{{\text{{number of revolutions}}}}\]

Plugging in the values, we get:

\[T = \frac{{25 \text{{ seconds}}}}{{14 \text{{ revolutions}}}}\]

Simplifying, we find:

\[T = \frac{{25}}{{14}} \text{{ seconds/revolution}}\]

Now, we can calculate the linear velocity:

\[v = \frac{{2 \pi \cdot 80 \text{{ cm}}}}{{\frac{{25}}{{14}} \text{{ seconds/revolution}}}}\]

Simplifying, we get:

\[v = \frac{{32 \pi}}{{5}} \text{{ cm/sec}}\]

Now that we have the linear velocity (\(v\)) and the radius (\(r\)), we can calculate the centripetal acceleration (\(a_c\)):

\[a_c = \frac{{(32 \pi/5)^2}}{{80}}\]

Simplifying, we find:

\[a_c = \frac{{1024 \pi^2}}{{200}}\]

Finally, the magnitude of the acceleration is the absolute value of \(a_c\):

\[|a_c| = \left|\frac{{1024 \pi^2}}{{200}}\right|\]

To find the direction of the acceleration, we can note that centripetal acceleration always points towards the center of the circle, in this case, towards the axis of rotation.

Therefore, the magnitude of the acceleration is \(\left|\frac{{1024 \pi^2}}{{200}}\right|\) and its direction is towards the center of the circle.