Question 1:

Two blocks, of masses m_1 and m_2,are connected by a model string, as shown below. Both blocks lie on a plane that is inclined to the horizontal at an angle α. The part of the plane supporting mass m_1 is smooth, so that there is no frictional force. The upper part of the plane is rough, and the coefficient of static friction between the block of mass m_2 and the plane is µ. The system is in equilibrium, and the string is taut

a) "Choose axes! Draw a diagram, marking clearly your choice of coordinate axes.

(b) "Draw force diagrams! Modelling the blocks as particles, draw two force diagrams showing all the forces acting on the two particles

(c) "Apply law(s)! Apply appropriate laws to find two vector equations, one scalar equation and one inequality.

(d) "Solve equation(s)! Assuming that the system remains in equilibrium, show that (m_1 + m_2)tanα ≤ µm_2.

(e) "Interpret solution! If the mass of the first blockism1=2kg,the coefficient of friction for the second block is µ = 2/ 3 and tanα = 1 /3,for what range of m_2 will the system remain in equilibrium?

Question 2:

A stone of mass m is thrown vertically up wards with speed u, and travels upwards under the in?uence of gravity and air resistance. Use the quadratic model of air resistance with the stone modelled as a sphere of effective diameter D.
This question follows Procedure1on page 197 of Unit 3, and the steps of this procedure appear in the parts of the question.

(a) "Draw picture! Draw a picture and mark on it any relevant information.

(b) "Choose axes! Choose an axis for this problem.

(c) "State assumptions! State any assumptions made.

(d) "Draw force diagram! Draw a force diagram.

(e) "Apply Newton's second law! Apply Newton's second law to obtain the equation dv/ dt = -k(v^2 + g/k), where v is the speed of the stone, x is the distance travelled by the stone, g is the magnitude of the acceleration due to gravity ,and k = c_2D^2/m is a constant.

(f) "Solve differential equation! Solve the differential equation and apply the initial condition to find the time t in terms of the speed v and the constants given above.

(g) "Interpret solution! Use your equation to show that the time take to reach the maximum height attained by the stone, t max ,is given by
T_max =1/√gkarctan√((k/g u)) .

This model predicts that as the initial velocity u increases ,there is an upper limit for the time to reach maximum height. Calculate this upper limit for a beach ball with diameter 0.5mand mass 0.1kg (use c_2 =0.2and g =9.81)