A spring ( k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41° with respect to the horizontal (Fig. 8–41). The spring is then released. Now the incline has a coefficient of kinetic friction μₖ. If the block, attached to the spring, is observed to stop just as it reaches the spring’s equilibrium position, what is the coefficient of friction μₖ?

검증된 단계별 안내

Identify the key concepts involved: The problem involves energy conservation, spring potential energy, kinetic energy, gravitational potential energy, and work done by friction. The spring is compressed, and the block moves up the incline, stopping at the spring's equilibrium position due to friction.

Write the energy conservation equation: The initial energy stored in the spring (spring potential energy) is converted into work done against friction and gravitational potential energy. The equation is: \( \frac{1}{2} k x^2 = m g h + f_k d \), where \( f_k \) is the frictional force, \( d \) is the distance traveled, and \( h \) is the height gained.

Express the frictional force: The frictional force is given by \( f_k = \mu_k N \), where \( N \) is the normal force. On an incline, \( N = m g \cos \theta \). Substitute \( f_k \) into the energy equation.

Relate the height \( h \) and distance \( d \): The height gained \( h \) is related to the distance traveled \( d \) along the incline by \( h = d \sin \theta \). Substitute this into the energy equation to eliminate \( h \).

Solve for \( \mu_k \): Rearrange the energy equation to isolate \( \mu_k \). The final expression will be \( \mu_k = \frac{\frac{1}{2} k x^2 - m g d \sin \theta}{m g d \cos \theta} \). Use the given values for \( k \), \( x \), \( m \), \( \theta \), and \( d \) (which is the spring's compression length, 0.50 m) to calculate \( \mu_k \).

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주요 개념

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

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this scenario, understanding how the spring's compression affects the force acting on the mass is crucial for analyzing the motion of the block on the incline.

Friction and Coefficient of Friction

Friction is the force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction (μₖ) quantifies this force and is defined as the ratio of the frictional force to the normal force. In this problem, determining μₖ is essential to understand how it affects the block's motion as it moves up the incline and eventually stops.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the potential energy stored in the compressed spring is converted into kinetic energy of the block and work done against friction as it moves up the incline. Analyzing these energy transformations is key to finding the coefficient of friction.