Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 10.2 m long. His speed at the bottom of the swing is 7.6 m/s. Tarzan does not know that the vine has a breaking strength of 1.0 103 N. What is the largest mass that Tarzan can have and still make it safely across the river?
At the bottom, Tarzan has a speed of
v = 7.6 m/s
This means that at that point he is actually accelerating upoward at the centripetal acceleration of:
a = v^2/r
where
r = 10.2 m
The centripetal acceleration arises because if you change direction the velocity vector changes, even if the speed itself doesn't change.
You then apply Newton's second law:
F = m a
The total force acting on Tarzan must be equal to his mass times his acceleration. If you take the upward direction as positive, then you can write this as:
F_vine - m g = m a
Where F_vine is the force exerted on Tarzan by the vine and m g is, of course the force exerted by the Earth's gravity field on Tarzan, which enters the equation with a minus sign because we've chosen the convention to take the upward direction as positive.
So, you see that:
F_vine = m (a + g)
Then, by Newton's third law, the force exerted by the Vine on Tarzan is minus the force exerted by Tarzan on the vine. Now the magnitude of this force can be 1.0 10^3 N at most. So, you can use this to solve for the maximum value for m.
thank you sooo much!
To determine the largest mass that Tarzan can have and still make it safely across the river, we need to calculate the tension in the vine when he swings at the bottom of the swing.
We can start by calculating the gravitational force acting on Tarzan at the bottom of the swing. The gravitational force (weight) can be calculated using the equation:
Weight = mass × acceleration due to gravity
The acceleration due to gravity on Earth is approximately 9.8 m/s^2. Let's assume Tarzan's mass is 'm'.
Weight = m × 9.8 m/s^2
Now, we can calculate the tension in the vine at the bottom of the swing using the centripetal force. The centripetal force can be calculated using the equation:
Centripetal Force = mass × velocity^2 / radius
Tarzan's velocity at the bottom of the swing is given as 7.6 m/s, and the radius is the length of the vine, which is 10.2 m. Thus, we have:
Centripetal Force = m × (7.6 m/s)^2 / 10.2 m
We know that the vine has a breaking strength of 1.0 × 10^3 N, so the tension in the vine should not exceed this value to ensure Tarzan's safety.
Tension ≤ Breaking Strength
m × 9.8 m/s^2 ≤ 1.0 × 10^3 N
Now, we can solve this inequality to find the maximum mass 'm' that Tarzan can have.
m ≤ (1.0 × 10^3 N) / (9.8 m/s^2)
m ≤ 102.04 kg
Therefore, Tarzan can have a maximum mass of 102.04 kg to safely make it across the river while swinging on the vine.