tarzan swings a vine of length 4m in a vertical circle under the influence of gravity. when the vine makes an angle of 20degree with the vertical, tarzan has a speed pf 5m/s. Find the centripetal acceleration, tangential acceleration and the resultant acceleration
been there , done that
To find the centripetal acceleration, tangential acceleration, and resultant acceleration, we can use the given information and some basic formulas.
First, let's define the variables:
- r: radius of the vertical circle (equals the length of the vine) = 4m
- θ: angle made by the vine with the vertical = 20 degrees
- v: speed of Tarzan = 5m/s
1. Centripetal Acceleration:
Centripetal acceleration is given by the formula:
ac = (v^2) / r
Here, v = 5m/s and r = 4m
ac = (5^2) / 4 = 25 / 4 = 6.25 m/s^2
Therefore, the centripetal acceleration is 6.25 m/s^2.
2. Tangential Acceleration:
Tarzan's speed remains constant, so the tangential acceleration is zero.
Therefore, the tangential acceleration is 0 m/s^2.
3. Resultant Acceleration:
The resultant acceleration is the vector sum of the centripetal and tangential accelerations.
Since the tangential acceleration is zero, the resultant acceleration is the same as the centripetal acceleration.
Therefore, the resultant acceleration is also 6.25 m/s^2.
To find the centripetal acceleration, tangential acceleration, and resultant acceleration, we can use the following equations:
1. Centripetal acceleration (ac):
ac = v^2 / r
2. Tangential acceleration (at):
at = α * r
3. Resultant acceleration (ar):
ar = √(ac^2 + at^2)
Here's how to calculate each of these values:
1. Centripetal acceleration (ac):
Given that Tarzan has a speed of 5 m/s and the radius of the circular motion can be calculated using trigonometry. When the vine makes an angle of 20 degrees with the vertical, it forms a right-angled triangle.
Using trigonometry: sin(20 degrees) = opposite/hypotenuse = r/4m
r = 4m * sin(20 degrees)
Now substitute the values into the equation to find ac:
ac = (5 m/s)^2 / (4m * sin(20 degrees))
2. Tangential acceleration (at):
The angle provided (20 degrees) doesn't give us enough information to directly calculate tangential acceleration. We would need the angular acceleration (α) to do so. So, without additional information about α, we cannot determine tangential acceleration accurately.
3. Resultant acceleration (ar):
Using the values calculated above for centripetal acceleration (ac) and assuming no tangential acceleration (at = 0), we can find the resultant acceleration.
ar = √(ac^2 + at^2) = √(ac^2 + 0^2) = ac
So, the centripetal acceleration (ac) will also be the resultant acceleration (ar) in this case. Note that if tangential acceleration (at) is provided, the resultant acceleration can be calculated using the formula mentioned above.