a) If the resultant of two equal forces is equal to the given force, find angle between the forces. [2] b) A ball is thrown vertically upwards at a rate of 40ms^ 1 . Find the time taken to attain…

a) Angle between two equal forces when their resultant equals one of them

Given:

• Two forces $F_1$ and $F_2$ are equal in magnitude: $F_1=F_2=F$.
• The resultant $R$ of these two forces is equal to one of the forces: $R=F$.

To find:

• The angle $\theta$ between the two forces.

Solution: When two forces act at a point, their resultant $R$ can be found using the law of parallelogram of forces (or vector addition). The magnitude of the resultant is given by: $$R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta }$$ Since $F_1=F_2=F$ and $R=F$, we substitute: $$F=\sqrt{F^2+F^2+2F\cdot F\cos \theta }$$ $$F=\sqrt{2F^2+2F^2\cos \theta }$$ Square both sides to eliminate the square root: $$F^2=2F^2+2F^2\cos \theta$$ Divide both sides by $F^2$ (since $F\ne 0$): $$1=2+2\cos \theta$$ $$2\cos \theta =1-2$$ $$2\cos \theta =-1$$ $$\cos \theta =-\frac{1}{2}$$ The angle $\theta$ whose cosine is $-\frac{1}{2}$ is: $$\theta =120^{\circ }$$

Answer: The angle between the two equal forces is $120^{\circ }$.

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b) Time taken to reach maximum height when a ball is thrown vertically upwards

Given:

• Initial velocity $u=40{\text{ms}}^{-1}$.
• Acceleration due to gravity $g=10{\text{ms}}^{-2}$ (acting downward).
• At maximum height, the final velocity $v=0$.

To find:

• Time taken to reach maximum height ($t$).

Solution: We use the equation of motion for uniformly accelerated motion: $$v=u-gt$$ At maximum height, $v=0$: $$0=40-10t$$ $$10t=40$$ $$t=\frac{40}{10}=4\text{seconds}$$

Answer: The time taken to attain the maximum height is 4 seconds.

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c) Mass of an object weighing 98 N on Earth

Given:

• Weight of the object $W=98\text{N}$.
• Acceleration due to gravity $g=9.8{\text{ms}}^{-2}$.

To find:

• Mass of the object ($m$).

Solution: Weight is related to mass and gravity by the formula: $$W=mg$$ Rearrange to solve for mass: $$m=\frac{W}{g}=\frac{98}{9.8}=10\text{kg}$$

Answer: The mass of the object is 10 kg.

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Key Concepts Recap

a) Resultant of Two Forces

• When two forces $F_1$ and $F_2$ act at an angle $\theta$, their resultant $R$ is given by: $$R=\sqrt{F_1^2+F_2^2+2F_1{F}_2\cos \theta }$$
• If $F_1=F_2=F$ and $R=F$, solving gives $\theta =120^{\circ }$.

b) Vertical Motion Under Gravity

• For an object thrown vertically upward:
  • Initial velocity: $u$.
  • Acceleration: $-g$ (since gravity acts downward).
  • At maximum height, velocity $v=0$.
  • Time to reach maximum height: $t=\frac{u}{g}$.

c) Weight and Mass

• Weight ($W$) is the force due to gravity: $W=mg$.
• Mass ($m$) is constant, while weight varies with gravity ($g$).

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