Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5–50). If his arms are capable of exerting a force of 1350 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 kg and the vine is 4.8 m long.

검증된 단계별 안내

Step 1: Identify the forces acting on Tarzan at the lowest point of his swing. At this point, the forces include the gravitational force (weight) acting downward and the tension in the vine acting upward. The net force provides the centripetal force required for circular motion.

Step 2: Write the equation for the net force at the lowest point of the swing. The tension in the vine (T) must counteract both the gravitational force and provide the centripetal force. The equation is:

T = mg + (mv² / r)

, where m is Tarzan's mass, g is the acceleration due to gravity, v is the speed at the lowest point, and r is the length of the vine.

Step 3: Substitute the known values into the equation. Tarzan's mass (m) is 78 kg, the gravitational acceleration (g) is approximately 9.8 m/s², and the length of the vine (r) is 4.8 m. The maximum tension Tarzan's arms can exert is 1350 N. The equation becomes:

1350 = (78)(9.8) + (78v² / 4.8)

Step 4: Rearrange the equation to solve for v². Subtract the gravitational force term from both sides:

1350 - (78)(9.8) = (78v² / 4.8)

. Then multiply through by 4.8 and divide by 78 to isolate v².

Step 5: Take the square root of the result to find the maximum speed v. The final expression for v is:

v = sqrt((1350 - (78)(9.8)) * (4.8 / 78))

. This gives the maximum speed Tarzan can tolerate at the lowest point of his swing.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Centripetal Force

Centripetal force is the net force that acts on an object moving in a circular path, directed towards the center of the circle. In Tarzan's case, as he swings from the vine, the tension in the vine must provide the necessary centripetal force to keep him moving in an arc. This force is crucial for determining the maximum speed he can tolerate at the lowest point of his swing.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle helps us calculate the forces acting on Tarzan at the lowest point of his swing, where the gravitational force and the tension in the vine interact to determine his maximum speed.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In Tarzan's swing, potential energy at the highest point converts to kinetic energy at the lowest point. Understanding this energy transformation is essential for calculating the maximum speed he can achieve during his swing.