Question

An air-track glider of mass 0.100 \rm kg is attached to the end of a horizontal air track by a spring with force constant 20.0 \rm N/m. Initially the spring is unstreched and the glider is moving at 1.50 \rm m/s to the right. With the air track turned off, the coefficient of kinetic friction is \mu_{\rm k}=0.47.

Part A How large would the coefficient of static friction \mu _{\rm{s}} have to be to keep the glider from springing back to the left when it stops instantaneously? Express your answer giving two significant figures.
Part B If the coefficient of static friction between the glider and the track is \mu_{\rm s} = 0.70, what is the maximum initial speed v_1 that the glider can be given and still remain at rest after it stops instantaneously? Express your answer giving two significant figures.

Answer 1

The question involves calculating the static friction coefficient necessary to prevent an air-track glider from moving when a spring stops it and finding the maximum initial velocity that the glider can have without overcoming static friction.

Step-by-step explanation:

The student's question involves the concepts of static and kinetic friction, oscillatory motion of masses on springs, and energy transformations in a physical system. The goal is to understand how frictional forces interact with the motion of objects attached to springs and how these forces can affect the maximum displacement of the spring or the object's stopping distance.

Part A

For the air-track glider to remain at rest when it stops, the maximum static friction force must be greater than or equal to the spring force at its maximum compression. Since the spring force can be calculated using Hooke's Law (Force = -kx where k is the spring constant and x is the displacement), we can set the static frictional force (μs * normal force) equal to the maximum force exerted by the spring to solve for the static coefficient.

Part B

If the coefficient of static friction between the glider and the track is 0.70, the maximum initial velocity can be found by equating the initial kinetic energy to the work done by friction (work is force times displacement) as the glider comes to rest. We would use the conservation of energy principle to equate the kinetic energy at the start with the potential energy at the point of maximum compression of the spring, considering the work done against friction.

Answer 2

The coefficient of static friction needed in Part A depends on the equalization of the maximum static friction force and spring force when the glider stops. For Part B, the maximum initial speed of the glider can be given by equating the initial kinetic energy to the work done by static friction using the given static friction coefficient of 0.70.

Step-by-step explanation:

To solve for the coefficient of static friction needed to prevent the glider from moving back (Part A), we first need to understand what forces are acting on the glider. When the glider stops, the spring force, which tends to pull it back to the left, must be equal to the maximum static friction force for it to remain at rest. The spring force at maximum compression or extension can be calculated using Hooke's Law (F = kx), where k is the spring constant (20.0 N/m) and x is the maximum displacement. This displacement is unknown, but we can infer from the condition that the glider stops instantaneously that we're interested in the initial stopping point. The maximum static friction force can be expressed as Fs_max = μ_s * N, where μ_s is the coefficient of static friction we wish to find and N is the normal force, which is equal to the weight of the glider for a horizontal surface.

For Part B, with a given coefficient of static friction of 0.70, the question is asking for the maximum initial speed at which the glider can move without overcoming the maximum static friction when it momentarily stops. To find this, we need to consider energy conservation and the work done by friction. The work done by the kinetic friction force (which opposes the glider's motion) will decrease the kinetic energy of the glider. The maximum amount of kinetic energy that can be converted to work done by static friction without causing motion is given by the equation W = Fs_max * x. Using this relationship, we can set the initial kinetic energy (1/2)*m*v_1^2) equal to the maximum work done by static friction to solve for v_1.