Forces and Equilibrium: Mathematics 9709 (Cambridge International AS & A Level)

Three coplanar forces act at a fixed point $O$ and hold it in equilibrium: a force of $9\text{ N}$ acting due east, a force of $12\text{ N}$ acting due north, and a third force of magnitude $F\text{ N}$.

What is the value of $F$?

A crate of mass $25\text{ kg}$ rests in equilibrium on rough horizontal ground. A rope attached to the crate is pulled with a constant tension of $80\text{ N}$, the rope making an angle of $30^\circ$ with the horizontal. The coefficient of friction between the crate and the ground is $0.4$. Take $g = 10\text{ m s}^{-2}$.

(a) Find the horizontal and vertical components of the $80\text{ N}$ tension. [2]

(b) By resolving forces perpendicular to the ground, find the normal reaction $R$ between the crate and the ground. [2]

(c) By resolving forces horizontally, find the frictional force acting on the crate, and state its direction. Hence show that the crate can indeed remain in equilibrium. [3]

A parcel of mass $8\text{ kg}$ rests on a rough plane inclined at $25^\circ$ to the horizontal. A light string, lying along a line of greatest slope, is attached to the parcel; the tension in the string acts up the plane. The coefficient of friction between the parcel and the plane is $0.2$. The parcel is in limiting equilibrium, on the point of sliding up the plane. Take $g = 10\text{ m s}^{-2}$.

(a) By resolving forces perpendicular to the plane, find the normal reaction $R$ between the parcel and the plane. [2]

(b) State the direction in which the frictional force acts, and hence find the tension $T$ in the string. [4]

Three coplanar forces act at a fixed point $O$ and hold it in equilibrium:

• a force of magnitude $40\text{ N}$ acting horizontally;
• a force of magnitude $30\text{ N}$ acting at $70^\circ$ to the horizontal, tilted to the same side as the $40\text{ N}$ force (so both forces have a horizontal component in the same direction);
• a third force of magnitude $F\text{ N}$.

Take the direction of the $40\text{ N}$ force as the positive $x$-direction, and "upward" (perpendicular to it) as the positive $y$-direction.

(a) Find the sum of the $x$-components and the sum of the $y$-components of the $40\text{ N}$ and $30\text{ N}$ forces. [3]

(b) Find the magnitude of $F$. [2]

(c) Find the angle that $F$ makes with the horizontal. [2]

A parcel of weight $45\text{ N}$ hangs at rest from one end of a light, inextensible string. The string passes over a smooth pulley fixed to the ceiling; the other end is held by a shop assistant, who pulls it so that this second part of the string is horizontal. The parcel remains in equilibrium.

What is the tension in the horizontal part of the string?

A rope is attached to a sledge and pulled with a force of $60\text{ N}$ at an angle of $35^\circ$ above the horizontal ground.

What is the vertical component of this force, correct to $3$ significant figures?

A decorative lamp of weight $20\text{ N}$ hangs in equilibrium, held by two light inextensible strings attached to two fixed points on a horizontal ceiling. The strings are on opposite sides of the lamp: one string makes an angle of $50^\circ$ with the ceiling and has tension $T_1$, and the other makes an angle of $35^\circ$ with the ceiling and has tension $T_2$.

(a) By resolving forces horizontally, write down an equation connecting $T_1$ and $T_2$. [2]

(b) By resolving forces vertically, write down a second equation connecting $T_1$ and $T_2$. [2]

(c) Solve your two equations simultaneously to find $T_1$ and $T_2$, giving each answer correct to $3$ significant figures. [3]

A crate of mass $30\text{ kg}$ rests in equilibrium on rough horizontal ground. A worker pushes the crate with a constant force of $100\text{ N}$, directed at $20^\circ$ below the horizontal (that is, pushing forwards and downwards into the ground). The coefficient of friction between the crate and the ground is $0.5$. Take $g = 10\text{ m s}^{-2}$.

(a) Find the horizontal and vertical components of the $100\text{ N}$ push. [2]

(b) By resolving forces perpendicular to the ground, find the normal reaction $R$ between the crate and the ground. [2]

(c) By resolving forces horizontally, find the frictional force required for equilibrium, find the maximum possible friction $\mu R$, and hence show that the crate can indeed remain in equilibrium. [3]

A small block rests on a rough plane inclined at an angle $\theta$ to the horizontal, held in place by friction alone (no other forces act on it). The coefficient of friction between the block and the plane is $\mu = 0.36$.

What is the maximum value of $\theta$, correct to $1$ decimal place, for which the block can remain in equilibrium on the plane without any additional force?

A block of mass $5\text{ kg}$ rests on a rough plane inclined at $32^\circ$ to the horizontal. The block is held in equilibrium by a horizontal force of magnitude $H$ newtons, directed so as to push the block towards the plane (its component along the plane acts up the slope). The coefficient of friction between the block and the plane is $0.3$. The block is in limiting equilibrium, on the point of sliding down the plane. Take $g = 10\text{ m s}^{-2}$.

(a) By resolving the weight and the force $H$ into components perpendicular to the plane, show that the normal reaction is $R = mg\cos32^\circ + H\sin32^\circ$. [3]

(b) State the direction in which friction acts, and by resolving forces along the plane, write down an equation connecting $H$, $\theta = 32^\circ$, the weight, and the limiting friction $\mu R$. [2]

(c) Combine your equations from parts (a) and (b) to find the value of $H$, correct to $3$ significant figures. [3]

(d) By Newton's third law, state the magnitude and direction of the force that the block exerts on the plane. [1]