A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long rope, and swings out over a lake (Fig. 8–49). He releases the rope when his velocity is zero. What is the angle θ when he releases the rope?
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Verified step by step guidance

1. Determine the initial kinetic energy of the student using the formula for kinetic energy: $K_i=\; (1/2)mv2$, where $m$ is the mass of the student (56 kg) and $v$ is the initial velocity (6.0 m/s).
2. Recognize that as the student swings upward, the kinetic energy is converted into gravitational potential energy. The gravitational potential energy is given by $U_f=\; mgh$, where $h$ is the height gained, $g$ is the acceleration due to gravity (9.8 m/s²), and $m$ is the mass of the student.
3. Apply the conservation of mechanical energy principle: $K_i+U_i=K_f+U_f$. Since the student starts at ground level, the initial potential energy $U_i$ is zero, and at the highest point, the final kinetic energy $K_f$ is zero. This simplifies to $K_i=U_f$, or $(1/2)mv2=\; mgh$.
4. Solve for the height $h$ using the equation $h=\; v2/(2g)$. Substitute the known values for $v$ (6.0 m/s) and $g$ (9.8 m/s²).
5. Relate the height $h$ to the angle $\theta$ using trigonometry. The rope forms a right triangle where the vertical height $h$ is opposite the angle $\theta$, and the rope length (10.0 m) is the hypotenuse. Use the equation $\cos \theta =\; (10.0\; -\; h)/10.0$ to solve for $\theta$.

Key Concepts

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the kinetic energy of the student running is converted into gravitational potential energy as he swings upward on the rope. At the highest point of the swing, when the student releases the rope, all kinetic energy is transformed into potential energy, allowing us to determine the angle at which he releases the rope.

Kinematics of Circular Motion

When the student swings on the rope, he undergoes circular motion. The kinematics of circular motion involves understanding how an object moves along a circular path, including concepts like centripetal acceleration and tension in the rope. The angle θ at which he releases the rope can be analyzed using the geometry of the circular path and the forces acting on the student at that point.

Trigonometric Relationships

Trigonometric relationships are essential for solving problems involving angles and distances in physics. In this case, the angle θ can be determined using trigonometric functions such as sine, cosine, or tangent, which relate the angle to the lengths of the sides of the triangle formed by the rope and the vertical line. Understanding these relationships allows for the calculation of the angle at which the student releases the rope.