Two particles P and Q of masses 5kg and 10kg respectively are connected by a light inextensible string over a small smooth pulley. P rests on an inclined plane and Q hangs vertical... Two particles P and Q of masses 5kg and 10kg respectively are connected by a light inextensible string over a small smooth pulley. P rests on an inclined plane and Q hangs vertically. The plane is inclined at an angle α where tan α = 0.75. The coefficient of friction between P and the plane is 0.2. The system is released from rest. (a) Find the acceleration of the system. (b) Find the tension in the string. Read more

Steps to Solve

1. Identify forces acting on particles P and Q

   For particle P (5 kg):

   • Weight: $W_P = m_P \cdot g = 5 \cdot 9.81 = 49.05 , \text{N}$
   • Normal force: $N = W_P \cdot \cos(\alpha)$
   • Frictional force: $F_f = \mu \cdot N = \mu \cdot (W_P \cdot \cos(\alpha))$

   For particle Q (10 kg):

   • Weight: $W_Q = m_Q \cdot g = 10 \cdot 9.81 = 98.1 , \text{N}$

2. Calculate the angle of inclination

   Given that $\tan(\alpha) = 0.75$, we can find $\alpha$ using: $$ \alpha = \tan^{-1}(0.75) \approx 36.87^\circ $$

3. Determine the normal force and frictional force on particle P

   • Calculate $N$: $$ N = W_P \cdot \cos(\alpha) = 49.05 \cdot \cos(36.87^\circ) $$
   • Calculate frictional force $F_f$: $$ F_f = \mu \cdot N = 0.2 \cdot N $$

4. Set up equations for motion

   For particle P:

   • Net force: $F_{\text{net}} = T - W_P \cdot \sin(\alpha) - F_f$

   For particle Q:

   • Net force: $F_{\text{net}} = W_Q - T$

5. Equate the net forces

   Since both particles are connected, the acceleration $a$ is the same:

   • For P: $$ m_P \cdot a = T - W_P \cdot \sin(\alpha) - F_f $$
   • For Q: $$ m_Q \cdot a = W_Q - T $$

6. Solve the equations simultaneously

   Rearrange and solve the system of equations to find acceleration $a$ and tension $T$.