A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is 0.050 m, and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is at its highest point.

검증된 단계별 안내

Identify the key parameters: mass of the cat (m) = 4.00 kg, amplitude (A) = 0.050 m, and the spring is at its natural length at the highest point.

At the highest point, the spring is unstretched, so the elastic potential energy (U_spring) is zero. Use the formula:

U_{spring} = \frac{1}{2} k x^{2}

, where x is the displacement from the natural length.

The kinetic energy (K) at the highest point is zero because the velocity is zero at this point. Use the formula:

K = \frac{1}{2} m v^{2}

, where v is the velocity.

Calculate the gravitational potential energy (U_gravity) at the highest point using the formula:

U_{gravity} = m g h

, where g is the acceleration due to gravity (9.81 m/s²) and h is the height above the lowest point, which is equal to the amplitude (0.050 m).

Sum the energies at the highest point:

E = U_{spring} + K + U_{gravity}

. Since Uspring and K are zero, the total energy is equal to the gravitational potential energy.

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주요 개념

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

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes the oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position. In this scenario, the cat attached to the spring exhibits SHM, characterized by its amplitude, frequency, and period, which are essential for calculating energy changes during the motion.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is compressed or stretched from its natural length. It is calculated using the formula U = 1/2 k x^2, where k is the spring constant and x is the displacement from the equilibrium position. At the highest point, the spring is unstretched, so the elastic potential energy is zero.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, calculated as U = mgh, where m is mass, g is acceleration due to gravity, and h is height above a reference point. In this problem, the reference point is the lowest point of the cat's motion, and the energy is highest at the top of the oscillation.