Is this right?
Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting 1350 N on the vine, what is his acceleration at the point where the vine makes an angle of 45 degrees from the vertical assuming maximum speed at this point. His mass is 78 kg and the vine is 5.2 m long.
I feel like I am missing something when I go through this but what I did was:
Tension would have to be 1350 right so
x direction: Fcx=Tsin(45) => 954.6N=Fcx
y direction: Fcy=Tcos(45)+Fg => 954.6 N - 764.4 N = 190.2 N = Fcy
(Fcx^2+Fcy^2)^0.5 = Fc = 973.4 N
973.4 N / 78 kg = 12.5 m/s/s inward
Your approach is almost correct, but there's a slight mistake in the calculation of the tension in the x-direction.
Let's go through the steps again:
1. Resolve the forces in the x and y directions:
In the x-direction:
The tension in the vine can be broken down into its x and y components. The x-component of the tension will help Tarzan accelerate tangentially along the arc. So, Fcx = T * sin(45°) = 1350 N * sin(45°) = 954.6 N.
In the y-direction:
The y-component of the tension provides the necessary force to balance out the gravitational force acting on Tarzan. So, Fcy = T * cos(45°) + mg = 1350 N * cos(45°) + (78 kg * 9.8 m/s^2) = 1255.02 N.
2. Find the net force:
To find the net force, we need to combine the x and y components by getting the vector sum of their squares:
Fc = sqrt(Fcx^2 + Fcy^2) = sqrt((954.6 N)^2 + (1255.02 N)^2) = 1563.63 N.
3. Calculate Tarzan's acceleration:
Finally, we can calculate Tarzan's acceleration using Newton's second law, F = ma:
Fc = m * a => 1563.63 N = 78 kg * a => a = 1563.63 N / 78 kg = 20.05 m/s^2.
Therefore, Tarzan's acceleration at the point where the vine makes an angle of 45 degrees is 20.05 m/s^2 inward.
It's great to see you applying the right formulas and attempting the problem step by step. Keep it up!