Problem 55 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. \(47 ) .\) If his arms are capable of exerting a force of 1350 \(\mathrm{N}\) on the rope, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 \(\mathrm{kg}\) and the vine is 5.2 \(\mathrm{m}\) long.

Step-by-Step Solution

Answer

The maximum speed is approximately 6.24 m/s.

Step 1: Identify Forces at Lowest Point

At the lowest point of the swing, Tarzan experiences two forces: the gravitational force (weight) acting downward and the tension in the vine acting upward. The net force at this point provides the centripetal force required for circular motion.

Step 2: Write the Equation for Net Force

The net force at the lowest point is given by:\[ T - mg = \frac{mv^2}{r} \]where \( T \) is the tension (1350 N), \( m \) is the mass (78 kg), \( g \) is the acceleration due to gravity (approximately 9.8 m/s²), \( v \) is the velocity, and \( r \) is the length of the vine (5.2 m).

Step 3: Calculate Gravitational Force

Calculate \( mg \), the gravitational force:\[ mg = 78 \times 9.8 = 764.4 \text{ N} \]

Step 4: Solve the Equation for Velocity

Rearrange the net force equation to solve for \( v^2 \):\[ v^2 = \frac{(T - mg) \cdot r}{m} \]Substituting the known values:\[ v^2 = \frac{(1350 - 764.4) \cdot 5.2}{78} \]\[ v^2 \approx \frac{585.6 \cdot 5.2}{78} \]\[ v^2 \approx 39.0 \]\[ v \approx \sqrt{39.0} \approx 6.24 \text{ m/s} \]

Step 5: Conclude with Maximum Speed

The maximum speed Tarzan can tolerate at the lowest point of his swing is approximately 6.24 m/s.

Key Concepts

Centripetal Force

When an object moves in a circular path, it changes direction continuously. This requires a force that is directed toward the center of the circle, known as centripetal force. For Tarzan swinging on a vine, the centripetal force is what keeps him moving in a circular arc instead of flying off in a straight line. At the lowest point in his swing, the net force acting on him combines his weight and the tension in the vine to provide this centripetal force. It is important to understand that centripetal force is not a new kind of force; it is the result of other forces, such as tension or gravity, acting on the object.

Tension in a String

Tension is the force conducted along a string, rope, or vine. It acts in a direction along the length of the wire and pulls equally on the objects on either end. For Tarzan, the tension in the vine acts upward against his weight. The tension reaches its maximum at the lowest point of the swing because this is when Tarzan is moving fastest and pulls hardest on the vine to keep him in circular motion. In this scenario, the tension in the vine is the maximum force Tarzan's arms can apply, calculated here as 1350 N. Understanding tension is crucial because it determines how strong the material holding the weight must be and how much force it can withstand before breaking.

Gravitational Force

Gravitational force is the force with which the Earth attracts a body toward its center. This force gives weight to physical objects and is calculated by multiplying the mass of the object by the acceleration due to gravity. In this context, the gravitational force acting on Tarzan is his weight, which is the product of his mass (78 kg) and the gravitational acceleration (approximately 9.8 m/s²). The gravitational force plays a critical role as it acts downward, working against the tension in the vine at the lowest point of the swing. Understanding gravitational force is essential, as it's a key component in many physical scenarios involving moving bodies and helps in calculating how other forces must act to result in the desired motion.

Velocity Calculation

Velocity is a vector quantity describing how fast an object is moving and in what direction. To find the maximum speed Tarzan can endure at the lowest point of his swing, we use the centripetal force equation that includes the tension in the vine and the gravitational force. The velocity equation derived is: \[ v^2 = \frac{(T - mg) \cdot r}{m} \]After inserting the values for tension, gravitational force, and the length of the vine, solving gives us that Tarzan's velocity at the lowest point is approximately 6.24 m/s. This calculation is crucial for ensuring safety, as exceeding this speed might result in excessive tension that could snap the vine or cause injury.