Question

A block of mass m slides across a horizontal frictionless surface at a speed of v0​. It then encounters a rough patch whose friction coefficient is not constant but increases according to

μ(x)=1−e⁻ˣ/ˣ⁰​

where x is measured from the transition (from frictionless to rough) and x0​ is some characteristic length that describes the scale at which the roughness of the surface changes.

(a) Use the work-KE theorem to determine an equation that relates the distance that the block slides before coming to rest to the initial speed. Note: It will be a transcendental equation that you will not be able to solve analytically.

Answer 1

The student's physics question relates to the work-energy principle, where the work of friction is calculated based on the varying coefficient of kinetic friction over a given distance, impacting the block's kinetic energy and stopping distance.

Step-by-step explanation:

The student's question involves applying the work-energy principle to determine the work done by friction and the stopping distance of a block moving across a surface with varying coefficients of friction.

By understanding that work done by friction (which is a non-conservative force) will change the kinetic energy of the block, we can relate the force of friction and the distance traveled to the initial kinetic energy of the block using the equation ∆K = W_{friction}.

For a block with mass m sliding on a horizontal surface, the work done by friction is equal to the integral of the frictional force over the distance traveled. The kinetic frictional force is μ(x)mg, where μ(x) is the position-dependent coefficient of kinetic friction and x is the distance the block has traveled since encountering the rough patch.

The work done by this frictional force as the block moves from an initial position to a final position can be calculated using the integral of μ(x)mg dx over the limits of the initial and final positions.