Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (see Fig). If his arms are capable of exerting a force of 1200 N on the rope, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 84 kg and the vine is 4.8 m long.

My physics optional extra work: This would be helpful so I can compare the solutions here as I do the work by myself.

To find the maximum speed Tarzan can tolerate at the lowest point of his swing, we need to consider the forces acting on him.

At the lowest point of the swing, the only force acting on Tarzan is his weight, which can be calculated using the formula:

Weight = mass × acceleration due to gravity

Using a standard value for acceleration due to gravity, which is approximately 9.8 m/s^2, we can calculate Tarzan's weight:

Weight = 84 kg × 9.8 m/s^2
Weight = 823.2 N

Since the total force exerted on the vine is the sum of the tension in the vine and his weight, we can write the equation:

Total force = tension + weight

Since the vine is at an angle to the vertical, we can break the tension into its vertical and horizontal components. The vertical component is equal to Tarzan's weight, while the horizontal component provides the centripetal force needed to keep Tarzan moving in a circular path.

The horizontal component of tension is given by:

Horizontal tension = tension × cos(angle)

The angle can be calculated using the length of the vine and the vertical height. The angle can be found using the equation:

cos(angle) = vertical height / length of the vine
cos(angle) = vertical height / 4.8 m

Given the height of the gorge is not provided, we will assume it is 0 meters, making the length of the vine equal to the vertical height. Therefore, the angle becomes:

cos(angle) = 0 / 4.8 m
cos(angle) = 0

As cos(angle) is equal to zero, it means the angle is 90 degrees (or π/2 radians). This implies that the vine is perpendicular to the vertical height.

Now, let's find the horizontal component of tension:

Horizontal tension = tension × cos(angle)
Horizontal tension = 1200 N × cos(90°)
Horizontal tension = 0 N

Since the horizontal tension is zero, there is no horizontal force acting and Tarzan will not move horizontally at the lowest point of the swing. Therefore, the maximum speed he can tolerate at the lowest point of his swing is zero meters per second.

Note: The above calculations assume there is no air resistance acting on Tarzan.

To tackle this problem, we need to apply the principles of circular motion and consider the forces acting on Tarzan during his swing. First, let's break down the problem and identify the forces involved.

1. The weight force, represented by the formula W = mg, where m is the mass of Tarzan (84 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). This force acts vertically downward.

2. The tension force provided by the vine when Tarzan is at the lowest point of his swing. This is the force Tarzan uses to swing, and it acts upward.

In this case, the maximum speed Tarzan can tolerate at the lowest point of his swing occurs when the tension force provided by the vine reaches its maximum value. This happens when the force Tarzan exerts on the vine is equal in magnitude to the weight force acting on him.

Now, let's calculate the tension force at the lowest point of Tarzan's swing:

1. Calculate the weight force:
W = mg = (84 kg)(9.8 m/s^2) = 823.2 N

2. At the lowest point of Tarzan's swing, the tension force and the weight force have equal magnitudes:
T = 823.2 N

Now, we'll use this information to find the maximum speed Tarzan can tolerate at the lowest point of his swing.

3. The centripetal force is provided by the tension force:
T = m * v^2 / r

Here:
T = 823.2 N (tension force)
m = 84 kg (mass of Tarzan)
v = ? (maximum speed Tarzan can tolerate at the lowest point)
r = 4.8 m (length of the vine)

4. Rearrange the formula to solve for v:
v^2 = T * r / m
v^2 = (823.2 N)(4.8 m) / 84 kg
v^2 = 46,630.4 N m / 84 kg

5. Take the square root of both sides to solve for v:
v = √(46,630.4 N m / 84 kg)

Now, you can substitute the values into this equation to find the maximum speed Tarzan can tolerate at the lowest point of his swing.

Please note that the final answer should include the proper unit of speed, which is m/s.

By following these steps, you should be able to compare your own solution to ensure accuracy.