Solving Friction Force and Acceleration Problems

[pttest]

Homework Statement

A block of mass m lies on a horizontal table. The coefficient of static friction between the block and the table is mu_s. The coefficient of kinetic friction is mu_k. (mu_k<mu_s)
a: Suppose you push horizontally with half the force needed to just make the block move. What is the magnitude of the friction force?
b: Suppose you push horizontally with precisely enough force to make the block start to move, and you continue to apply the same amount of force even after it starts moving. Find the acceleration a of the block after it begins to move? {Express your answer in terms of some or all of the variables mu_s, mu_k, and m, as well as the acceleration due to gravity g}

Homework Equations

F_k = mu_k.N (N is normal force)
F_s = mu_s.N

The Attempt at a Solution

I was wondering a: 1/2.mu_k.m.g
b: mu_k.g
could anybody please expalin this problem?

Thanks in advance

 

[rl.bhat]

With in the limit, the frictional force is a self adjusting force.
So in a: frictional force is f/2, because the block is not moving.
In b: Net force acting on the block is f - μk*mg. Now find the acceleration.

 

[nlightner]

I would first like to clarify that the equations you have listed are correct. However, the equations you have provided are for calculating the friction force when the block is already in motion. To solve this problem, we need to consider the different scenarios outlined in the homework statement.

a) In this scenario, the block is not moving and we are pushing with half the force needed to make it move. In this case, the block is in equilibrium and the forces acting on it are balanced. This means that the friction force (F) must be equal to half the force we are pushing with (F/2). Therefore, we can set up the following equation: F = F/2 = mu_s*N. We can rearrange this equation to solve for the friction force (F): F = 2*mu_s*N. We know that the normal force (N) is equal to the weight of the block, which is mg. Therefore, the friction force in this scenario is 2*mu_s*mg.

b) In this scenario, we are pushing with just enough force to make the block start moving and then we continue to apply the same amount of force. In this case, the block will experience a net force in the direction of the push, which will cause it to accelerate. To find the acceleration (a), we can use Newton's second law, which states that the net force (F_net) is equal to the mass (m) times the acceleration (a): F_net = m*a. We know that the net force is equal to the force we are pushing with (F) minus the friction force (F): F_net = F - F = 0. Therefore, we can set up the following equation: F = mu_k*N = mu_k*mg = m*a. We can rearrange this equation to solve for the acceleration (a): a = mu_k*g. This means that the acceleration of the block in this scenario is equal to mu_k times the acceleration due to gravity (g).

I hope this helps to clarify the problem and the solutions. It is important to carefully consider the forces at play and the different scenarios outlined in the homework statement in order to arrive at the correct solutions.