Piles of snow on slippery roofs can become dangerous projectiles as they melt. Consider a chunk of snow at the ridge of a roof with a slope of 24°. As the snow begins to melt, the coefficient of static friction decreases and the snow finally slips. Assuming that the distance from a chunk of snow to the edge of the roof is 6.0 m and the coefficient of kinetic friction is 0.20, calculate the speed of the snow chunk when it slides off the roof.

검증된 단계별 안내

Step 1: Identify the forces acting on the snow chunk. The forces include the gravitational force (mg), the normal force (N), and the kinetic frictional force (f_k). The gravitational force can be broken into two components: one parallel to the slope (mg sin(θ)) and one perpendicular to the slope (mg cos(θ)).

Step 2: Write the equation for the net force along the slope. The net force (F_net) is the difference between the downslope gravitational force (mg sin(θ)) and the kinetic frictional force (f_k). The kinetic frictional force is given by f_k = μ_k * N, where N = mg cos(θ). Thus, F_net = mg sin(θ) - μ_k * mg cos(θ).

Step 3: Use Newton's second law to find the acceleration of the snow chunk. According to Newton's second law, F_net = ma, where a is the acceleration. Substituting the expression for F_net, we get a = g(sin(θ) - μ_k * cos(θ)).

Step 4: Use kinematic equations to calculate the final speed of the snow chunk. The initial velocity (v_0) is 0 since the snow starts from rest. The distance traveled along the slope is 6.0 m. Using the kinematic equation v^2 = v_0^2 + 2a * d, substitute v_0 = 0, a = g(sin(θ) - μ_k * cos(θ)), and d = 6.0 m to find the final velocity v.

Step 5: Simplify the expression for the final velocity. After substituting the known values for g (9.8 m/s^2), θ (24°), μ_k (0.20), and d (6.0 m), calculate the final velocity v. Ensure that the trigonometric functions sin(θ) and cos(θ) are evaluated correctly.

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주요 개념

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

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. It is categorized into static friction, which prevents motion, and kinetic friction, which acts when objects are sliding past each other. The coefficient of friction quantifies this force, with values depending on the materials involved. In this scenario, the transition from static to kinetic friction is crucial as the snow chunk begins to slide off the roof.

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, expressed as F = ma. This principle is essential for analyzing the forces acting on the snow chunk as it slides down the roof. The gravitational force and frictional force will determine the net force and, consequently, the acceleration of the snow chunk.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations will be used to calculate the final speed of the snow chunk as it slides off the roof, taking into account the distance traveled and the acceleration due to gravity and friction.