1. A block of mass m is suspended by two strings as shown. Find the tension in the horizontal string as a function of m and .

Fnet,x = T1 T2 cos = 0Fnet,y = T2 sin mg = 0From the second equation T2 mg/ sin Substituting: T1 = mg cot

2. Two blocks are attached by a string over a massless pulley as shown. The coefficient of friction on the slope is = 0.3. The mass M= 8 kg, and the mass m = 3 kg. The slope angle is = 350. Find (a) the acceleration of the mass M (b) the speed of mass M when it hits the ground if it is released at rest from a height h = 2 m.

For mass M, Mg T = MaFor mass m, T f mg sin = ma N mg cos fFrom the last two equations, f = mg cos

Substituting: T mg cos mg sin = maAdding the first equation: Mg mg cos mg sin = ma + Maa = (Mg mg cos mg sin )/(m+M)

To find the time to hit the ground, use v2 = 2ah where a is the acceleration found above.

Version 1: = 0.3, M= 8 kg, m = 3 kg, h = 2 m a = 4.94 m/s2 v= 4.44 m/sVersion 2: = 0.2, M= 8 kg, m = 1 kg, h = 2 m a =7.91 m/s2 v= 5.62 m/sVersion 3: = 0.1, M= 7 kg, m = 2 kg, h = 3 m. a = 6.19 m/s2 v= 6.09 m/sVersion 4: = 0.2, M= 6 kg, m = 2 kg, h = 1 m. a = 5.54 m/s2 v= 3.33 m/s 3. A box of mass m = 10.0 kg slides 4.0 m down the ramp shown above, then collides with a spring whose spring constant is k = 220 N/m. The coefficient of friction is 0.2. Find the maximum compression of the spring.Suppose the compression is x meters.The change in gravitational PE = - mg (4.0+ x) sin The change in KE = 0 (when the block comes to rest)Change in elastic PE = kx2 .Force of friction = mg cos .Work done by friction = - mg cos (4.0+ x).Hence mg (4.0+ x) sin kx2 = mg cos (4.0+ x).i.e. kx2 + x ( mg cos mg sin ) + 4.0 ( mg cos mg sin ) =0This is a quadratic equation, and has two solutionsx= -(b/2a) ( (b2 4ac))/(2a)where a = kb = ( mg cos mg sin )c = 4.0 ( mg cos mg sin )Since x is positive, only the positive solution is to be kept i.e.x= -(b/2a) + ( (b2 4ac))/(2a)

Version 1: m = 10.0 kg, k = 220 N/m, 0.2. x = 1.23 mVersion 2: m = 10.0 kg, k = 190 N/m, 0.1. x = 1.54 mVersion 3: m = 10.0 kg, k = 210 N/m. 0.1. x = 1.45 mVersion 4: m = 10.0 kg, k = 210 N/m. 0.2. x = 1.27 m