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1. Two particles $A$, of mass 2kg, and $B$, of mass 5 kg, are connected via a light inextensible string. The string passes over a smooth pulley which is at the top of a rough inclined plane, angled at 30° to the horizontal. The coefficient of friction between $A$, which lies on the plane, and the plane is $\dfrac{\sqrt{3}}{2}$. $B$ hangs freely off the edge of the plane.
Calculate the tension in the string when the particles are released from rest.
2. A car of mass 2800 kg and its trailer of mass 1200 kg are travelling on a horizontal road and are attached via a light inextensible rod which is angled at 20° to the horizontal, with the car end being higher.
The car accelerates at 0.125 m s-2, and there is a constant resistance to motion of 600 N on the car and 270 N on the trailer.
Find the tension in the rod.
3. A box, $B$, of mass 7kg and a particle, $A$, of mass 3kg are attached to a light inextensible string. The box rests on a rough plane inclined at an angle $\theta$ to the horitontal, where $\tan\theta = \dfrac{3}{4}$. The coefficient of friction between the box and the plane is $0.6$. The string passes over a smooth pulley at the bottom of the plane, and the particle hangs off the edge.
Find the acceleration of the system after it is released from rest.
4. A trailer of mass 600 kg is connected to a car of mass 1500 kg by a light inextensible rod. The car is climbing a a hill, which is inclined at an angle of $\theta$ to the horizontal, where $\sin\theta = \dfrac{7}{25}$. There is a constant resistance to the motion parallel to the hill of 400 N on the car and 300 N on the trailer. The car produces a driving force of 8400 N. Find the acceleration of the car.
5. Two particles $A$, with mass 5kg, and $B$, with mass 1kg, are connected via a light inextensible string. The string passes over a smooth pulley which is at the top of a rough inclined plane. $A$ rests on the plane and $B$ hangs off the edge. The plane is inclined at an angle $\theta$ to the horizontal, where $\tan\theta = \dfrac{3}{4}$. The coefficient of friction between $A$ and the plane is $0.2$.
Find the tension in the string when the system is released from rest.
6. Two particles $A$ and $B$ have masses 0.5 kg and 0.2 kg respectively. The particles are attached to ends of a light inextensible string, as shown below. $A$ lies on a smooth incline plane, which is at an angle of $\theta$ to the horizontal, where $\tan\theta = 0.75$. $B$ lies on a rough horizontal table.
When the particles are released from rest, $A$ moves 2.25 m down the plane in the first 1.5 seconds of motion. Find, as a fraction, the coefficient of friction between $B$ and the table.
7. Two particles $A$, of mass 2 kg, and $B$, of mass 5 kg, are connected via a light inextensible string. The string passes over a smooth pulley which is at the top of a smooth inclined plane, angled at 30° to the horizontal. $A$ lies on the plane and $B$ hangs freely off the edge of the plane.
Show that the force exerted by the string on the pulley can be written in the form $a\sqrt{3}$ N.
8. Two particles $A$ and $B$ have masses 3 kg and 2 kg respectively and are joined by a light inextensible string which passes over a smooth pulley, as shown below. $A$ is at rest on a rough incline plane angled at $\theta$ to the horizontal, where $\tan\theta = 0.75$, and $B$ is at rest on a rough horizontal table. The coefficients of friction between $A$ and the plane and between $B$ and the table are both $\dfrac{1}{7}$.
A constant force of 30 N is applied to $B$ away from the pulley. After 1.5 seconds, the string breaks. Find the total distance $A$ travels up the plane.
9. Two particles $A$, of mass 4.5 kg, and $B$, of mass 0.4 kg, are connected via a light inextensible string. The string passes over a smooth pulley which is at the top of a rough inclined plane, angled at $\theta$ to the horizontal where $\tan\theta = 0.75$. $A$ lies on the plane and the coefficient of friction between $A$ and the plane is $0.5$. $B$ hangs freely off the edge of the plane.
After the particles are released from rest, find the force exerted by the string on the pulley.
10. Two particles, $A$ and $B$, of mass $m$ kg and $3$ kg respectively, are connected by a light inextensible string. $A$ is held on a smooth plane angled at 30° to the horizontal. The string passes over a smooth pulley at the top of the plane, and $B$ hangs freely from the pulley, 0.25 m above the ground.
The system is released from rest and $B$ accelerates towards the ground at $3.92$ m s-2. After $B$ lands, it does not rebound. Find, as a fraction of a second, the time between $B$ landing on the ground and the instant when $A$ reaches its highest point.
11. $A$ and $B$ are two particles on smooth slopes as shown below, connected via a light inextensible string which passes over a pulley at the top of the slopes. $A$ sits on a slope angled at 30° to the horizontal and has mass $m$ kg. $B$ sits on a slope angled at 45° to the horizontal and has mass $2$ kg.
When the system is released from rest, $B$ accelerates down the slope at $0.5$ m s-2.
Show that $m = \dfrac{a + b\sqrt{2}}{27}$ kg, where $a$ and $b$ are constants to be found.
12. $A$ and $B$ are two particles on rough slopes as shown below, connected via a light inextensible string which passes over a pulley at the top of the slopes. Both slopes are angled at 30° to the horizontal. $A$ has mass 2 kg and $B$ has mass $m$ kg. The coefficient of friction between $A$ and the slope is $0.2$ and the coefficient of friction between $B$ and the slope is $0.4$.
Given that the system is in equilibrium, show that that the maximum value of $m$ is $\dfrac{a + b\sqrt{3}}{13}$, where $a$ and $b$ are constants to be found.