Question

A block of mass m is at rest at the origin at t = 0. It is pushed with constant force F₀ from 𝓍 = 0 to 𝓍 = L across a horizontal surface whose coefficient of kinetic friction is μₖ = μ₀ ( 1 - 𝓍/L ) . That is, the coefficient of friction decreases from μ₀ at 𝓍 = 0 to zero at 𝓍 = L. b. Find an expression for the block's speed as it reaches position L.

Accepted Answer

Step 1: Begin by identifying the forces acting on the block. The block experiences a constant applied force F₀, a frictional force that depends on position 𝓍, and the normal force balancing the gravitational force. The frictional force is given by F_friction = μₖ * m * g, where μₖ = μ₀ * (1 - 𝓍/L). Step 2: Write the net force acting on the block as a function of position 𝓍. The net force is F_net = F₀ - F_friction. Substituting the expression for F_friction, we get F_net = F₀ - μ₀ * (1 - 𝓍/L) * m * g. Step 3: Use Newton's second law to relate the net force to the block's acceleration. Newton's second law states F_net = m * a, where a is the acceleration. Substituting F_net, we have m * a = F₀ - μ₀ * (1 - 𝓍/L) * m * g. Simplify to find the acceleration: a = (F₀/m) - μ₀ * g * (1 - 𝓍/L). Step 4: To find the block's speed at position 𝓍 = L, use the work-energy principle. The work done by the net force is equal to the change in kinetic energy. The work done by the applied force and friction can be calculated by integrating the net force over the distance from 𝓍 = 0 to 𝓍 = L. The work-energy principle states: W_net = ΔK = (1/2) * m * v² - 0, where v is the speed at 𝓍 = L. Step 5: Perform the integration to find the work done by the net force. The net force is F_net = F₀ - μ₀ * (1 - 𝓍/L) * m * g. Integrate F_net with respect to 𝓍 from 0 to L: W_net = ∫[0 to L] (F₀ - μ₀ * (1 - 𝓍/L) * m * g) d𝓍. Solve this integral to find W_net, then equate it to (1/2) * m * v² to solve for v, the block's speed at position L.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Newton's Second Law of Motion

Friction and its Coefficient

Work-Energy Principle