tarzan swings a vine of length 4m in a vertical circle under the influence of gravity. When the vine makes an angle of 20degrees with the vertical, Tarzan has a speed of 5m/s. Find centripetal acceleration, tangential acceleration and the resultant acceleration.
Explain the answer properly.
Height of Tarzan above hanging straight down = 4(1-cos 20)= .241 meters
v = 5
Ac = v^2/R = 25/4 = 6.25 m/s^2
which is radial acceleration
weight = m g = 9.81 m
tangential component of weight
= m g sin 20 = 3.36 m
so
tangential acceleration = 3.36 m/s^2
toward equilibrium point
resultant a = sqrt (3.36^2 + 6.25^2)
Well, let's swing into action and solve this Tarzan-themed physics problem!
First, let's find the centripetal acceleration. In this case, centripetal acceleration is the acceleration directed toward the center of the circle. We can calculate it using the formula for centripetal acceleration:
ac = v^2 / r
where ac is the centripetal acceleration, v is the speed of Tarzan, and r is the radius of the circle.
Given that Tarzan's speed is 5 m/s and the radius of the circle is 4 m, we can plug these values into the formula:
ac = (5 m/s)^2 / 4 m
Simplifying this equation gives us:
ac = 25 m^2/s^2 / 4 m
ac = 6.25 m/s^2
So, the centripetal acceleration of Tarzan swinging is 6.25 m/s^2.
Next, let's calculate the tangential acceleration. Tangential acceleration is the acceleration tangent to the circle at any given point. In this case, we don't have enough information to calculate it directly. However, we can determine that at the highest point of Tarzan's swing (when the vine makes an angle of 20 degrees with the vertical), the tangential acceleration is zero. This is because at that point, Tarzan's velocity is purely horizontal, perpendicular to the tangent.
Finally, the resultant acceleration is the combination of centripetal acceleration and tangential acceleration. Since the tangential acceleration is zero at the highest point, the resultant acceleration is equal to the centripetal acceleration at that specific moment.
Therefore, the centripetal acceleration is 6.25 m/s^2, the tangential acceleration is 0 m/s^2, and the resultant acceleration is 6.25 m/s^2.
I hope that explanation didn't make you vine-d yourself to sleep!
To find the centripetal acceleration, tangential acceleration, and resultant acceleration of Tarzan swinging on a vine, we need to break down the acceleration components.
1. Centripetal Acceleration:
Centripetal acceleration is the acceleration directed towards the center of a circular path. In this case, the vine swings Tarzan in a vertical circle. When the vine makes an angle of 20 degrees with the vertical, we can use trigonometry to find the vertical component of the centripetal acceleration.
The centripetal acceleration can be calculated as:
Centripetal acceleration = (velocity^2) / radius
Given:
Velocity (v) = 5 m/s
Radius (r) = 4 m
Centripetal acceleration = (5^2) / 4
Centripetal acceleration = 25 / 4
Centripetal acceleration = 6.25 m/s^2
2. Tangential Acceleration:
Tangential acceleration is the linear acceleration along the tangent of the circular path. In this case, Tarzan's speed is constant, so there is no tangential acceleration. Therefore, the tangential acceleration is 0 m/s^2.
3. Resultant Acceleration:
To find the resultant acceleration, we need to consider both the centripetal acceleration and tangential acceleration (or lack thereof). Since tangential acceleration is 0 m/s^2, the resultant acceleration is equal to the centripetal acceleration.
Resultant acceleration = Centripetal acceleration = 6.25 m/s^2
Therefore, the centripetal acceleration is 6.25 m/s^2, the tangential acceleration is 0 m/s^2, and the resultant acceleration is also 6.25 m/s^2.
To find the centripetal acceleration, tangential acceleration, and resultant acceleration, we need to understand their definitions and formulas.
1. Centripetal Acceleration:
Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It constantly changes the direction of the object towards the center of the circle. The formula for centripetal acceleration is given by:
a_c = (v^2) / r,
where a_c is the centripetal acceleration, v is the velocity, and r is the radius of the circular path.
2. Tangential Acceleration:
Tangential acceleration is the acceleration in the direction tangent to the circular path. It is caused by changes in speed. The formula for tangential acceleration is:
a_t = dv / dt,
where a_t is the tangential acceleration, dv is the change in velocity, and dt is the change in time.
3. Resultant Acceleration:
Resultant acceleration is the overall acceleration experienced by the object, considering both the centripetal and tangential accelerations. It can be calculated using the Pythagorean theorem:
a_r = sqrt((a_c)^2 + (a_t)^2),
where a_r is the resultant acceleration.
Now, let's apply these concepts to solve the problem.
Given:
Length of the vine (radius), r = 4m
Angle with the vertical, θ = 20 degrees
Speed, v = 5m/s
1. Centripetal Acceleration:
To find the centripetal acceleration, we need to calculate the radius of the circular path. Since Tarzan swings from a vine, the length of the vine (4m) is the radius of the circle. Therefore, r = 4m.
Using the formula: a_c = (v^2) / r,
Substituting the given values, we get: a_c = (5^2) / 4 m/s^2
Simplifying, we find: a_c = 25 / 4 m/s^2
2. Tangential Acceleration:
In this case, there is no information provided on the change in velocity or time. So, we assume that the tangential acceleration is zero.
3. Resultant Acceleration:
Using the formula: a_r = sqrt((a_c)^2 + (a_t)^2),
Since a_t = 0 (no tangential acceleration), the equation becomes: a_r = sqrt((a_c)^2 + 0)
Simplyfying, we find: a_r = sqrt((a_c)^2)
Thus, a_r = |a_c| (taking the positive value because acceleration is always positive).
Therefore, the centripetal acceleration is 25/4 m/s^2, the tangential acceleration is 0 m/s^2, and the resultant acceleration is 25/4 m/s^2.