Springs and Friction. Two net force problems are required in the solution of this problem.

A note about spring problems: When doing net force problems with a spring that obeys Hooke's Law, place the origin of your coordinate system at the unstretched position of the spring. Then you can substitute T = k|x| for the spring force. Note that this is a positive quantity, consistent with the practice of substituting force magnitudes into net force equations. You decide on the sign according to whether the force points in the direction of or opposite to the direction selected for +x (or +y).

1. A spring of spring constant k is attached to a block of mass m which initially rests on a horizontal table. A horizontal force pulls on the spring to the left. The coefficients of kinetic and static friction between the block and the table are μ_k and μ_s respectively. See the diagram to the right.

1. The force on the spring is increased from 0 to a magnitude T_s that is just enough to start the block sliding. As the force is applied, the spring stretches from 0 to an amount x_s. Determine an expression for x_s in terms of m, k, g, and μ_s only.

2. Do the following checks on your equation from part a.

   1. Units: Substitute in the units of the quantities on the right-hand side to see if they reduce to the units of x_s.

   2. Signs: Check whether the signs of the quantities on the right-hand side of the equation result in the correct sign for x_s. Tell how you know by giving the sign of each quantity.

3. The tension force is increased to a value greater than T_s so that the block slips. Then the force is quickly adjusted to a value T_k so that the block slides with constant velocity. Determine an expression for x_k in terms of m, k, g, and μ_k only.

4. Which is larger, x_k or x_s? Tell how you know.

5. If the tension force is decreased to a value less than T_k, tell what will happen to the block. Explain in terms of force and acceleration.

Connected Objects. Three force diagrams and four net force equations are required for the solution of the following problem. Use appropriate subscripts.

2. Three blocks are connected by strings in the arrangement shown in the figure to the right. The table and the pulley are frictionless. Ignore the masses of the strings and pulley. Assume the strings are taut and do not stretch.

   1. The system accelerates to the right and down. Determine an expression for the acceleration in terms of the masses of the blocks and g.

   2. Determine expressions for the tension forces in the three sections of string in terms of the three masses and g only.

   3. Check your results for the special case of m₂ = 0 (middle block). Show what your equations reduce to, tell what you expect, and tell whether the results agree with your expectations.

Tension and Friction in 2 Dimensions. There's just one net force problem here, but don't get tripped up on the algebra.

3. A rope attached to the front of a box of mass m is pulled at an angle θ with force T as shown to the right. Let a_x represent the resulting acceleration of the box. The coefficient of kinetic friction between the box and the surface is μ_k.

   1. Determine an equation for T in terms of θ, a_x, m, g, and μ_k.

   2. Check your equation for T by substituting θ = 0°. Is the result expected? Explain.

Friction on an Incline. Both kinetic and static friction are involved in this problem.

4. A coin of mass m is initially given a push to start it moving up a plane inclined at an angle θ  = 20° with the horizontal. If the plane is long enough, the coin comes to a stop. Given that μ_s = 0.40, determine if there is sufficient static friction to prevent the coin from sliding back down. Clearly show how you determined your answer.

Circular Motion. Here's a hint: This is the most common circular motion problem.

5. To study circular motion, a student uses the hand-held device shown to the right, which consists of a rod on which a spring scale is attached. A polished glass tube attached at the top serves as a guide for a light cord attached to the spring scale. A ball of mass m = 0.25 kg is attached to the other end of the cord. The student swings the cord around at constant frequency in a horizontal circle with radius r = 0.50 m.  The period of the ball’s motion is 0.75 s. Assume friction and air resistance are negligible.

   1. Determine the magnitude and direction of the acceleration of the ball.

   2. The student finds that, try as she might, she can’t keep the cord horizontal. Draw a force diagram for the ball and write a net force equation to use in explaining why it isn’t possible for the cord to remain exactly horizontal. The diagram and equation should support an explanation in sentence form.

   3. Determine the angle that the cord makes with the horizontal.