A simple pendulum of length 2.00 m is made with a mass of 2.00 kg. The mass has a speed of 3.00 m/s when the pendulum is 30.0° above its lowest position.

What is the maximum angle (in degrees) away from the lowest position the pendulum will reach?

Question

A simple pendulum of length 2.00 m is made with a mass of 2.00 kg.  The mass has a speed of 3.00 m/s when the pendulum is 30.0° above its lowest position.

What is the maximum angle (in degrees) away from the lowest position the pendulum will reach?

BlackTom AI · Accepted Answer

Question restatement: A simple pendulum of length 2.00 m and mass 2.00 kg has a speed of 3.00 m/s when it is 30.0° above the lowest position. Find the maximum angle away from the lowest position.

Option-by-option analysis:
Option 1: 50.5
- This value matches the energy conservation calculation. At the 30° position, the pendulum has both kinetic energy (1/2 m v^2) and potential energy relative to the lowest point (m g h, with h = L(1 − cos 30°)). The total mechanical energy is conserved, so at the turning point (max angle), kinetic energy is zero and m g L(1 − cos θ_max) equals the total energy. Solving yields θ_max ≈ 50.5°, which is consistent with the computed cosine value derived from the energy equation. This option is correct.

Option 2: none of these
- This would be chosen if none of the listed angles could satisfy the energy balance. However, performing the energy calculation shows a valid angle around 50.5°, so this option is incorrect.

Option 3: 62.8
- If θ_max were 62.8°, we would have cos θ_max ≈ cos 62.8° ≈ 0.460. Then 1 − cos θ_max ≈ 0.540, and m g L(1 − cos θ_max) would be about 2 × 9.8 × 2 × 0.540 ≈ 21.1 J, which does not match the total energy (≈14.25 J) found from the initial conditions. Therefore this angle would not conserve energy given the stated initial speed, making this option incorrect.

Option 4: 33.3
- For θ_max = 33.3°, cos θ_max ≈ cos 33.3° ≈ 0.834. Then 1 − cos θ_max ≈ 0.166, and m g L(1 − cos θ_max) ≈ 2 × 9.8 × 2 × 0.166 ≈ 6.5 J, far short of the total energy (≈14.25 J). This would imply leftover kinetic energy at the turning point, which is impossible for a true maximum angle, so this option is incorrect.

Option 5: 72
- With θ_max = 72°, cos ≈ 0.309, so 1 − cos ≈ 0.691. The corresponding energy would be m g L(1 − cos θ_max) ≈ 2 × 9.8 × 2 × 0.691 ≈ 27.1 J, which again overshoots the total energy available from the initial conditions. Hence this option is incorrect.

Summary: By applying energy conservation, the total mechanical energy at the starting position (30° with velocity 3 m/s) equals the potential energy at the maximum angle (where velocity is zero). Only 50.5° satisfies this condition, while the other angles yield energies inconsistent with the given initial state.

A simple pendulum of length 2.00 m is made with a mass of 2.00 kg. The mass has a speed of 3.00 m/s when the pendulum is 30.0° above its lowest position. What is the maximum angle (in degrees) away from the lowest position the pendulum will reach?单项选择题

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