Force: Mathematical problems & solutions-4

Problem 1

Problem: A 3 kg object is subjected to a force that accelerates it at 4 m/s². Calculate the force applied. Solution: $𝐹=𝑚𝑎=3\text{ kg}\times 4{\text{ m/s}}^2=12\text{ N}$ The force applied is $12\text{ N}$.

Problem 2

Problem: A 15 kg box is being pushed across a horizontal surface with a coefficient of kinetic friction of 0.25. Calculate the force required to keep the box moving at a constant velocity. Solution: ${𝐹}_{𝑁}=𝑚𝑔=15\text{ kg}\times 9.8{\text{ m/s}}^2=147\text{ N}$ ${𝐹}_{𝑓}=𝜇{𝐹}_{𝑁}=0.25\times 147\text{ N}=36.75\text{ N}$ The force required is $36.75\text{ N}$.

Problem 3

Problem: A 4 kg object is whirled in a horizontal circle of radius 2 meters at a speed of 3 m/s. Calculate the centripetal force. Solution: ${𝐹}_{𝑐}=\frac{𝑚{𝑣}^2}{𝑟}=\frac{4\text{ kg}\times (3\text{ m/s}{)}^2}{2\text{ m}}=18\text{ N}$ The centripetal force is $18\text{ N}$.

Problem 4

Problem: A spring with a spring constant of 300 N/m is compressed by 0.1 meters. Calculate the force exerted by the spring. Solution: $𝐹=-𝑘𝑥=-300\text{ N/m}\times 0.1\text{ m}=-30\text{ N}$ The force exerted is $30\text{ N}$.

Problem 5

Problem: A 5 kg ball moving at 12 m/s is brought to rest in 3 seconds by a constant force. Calculate the force. Solution: $\mathrm{\Delta }𝑝=𝑚({𝑣}_{𝑓}-{𝑣}_{𝑖})=5\text{ kg}\times (0-12\text{ m/s})=-60\text{ kg m/s}$ $𝐹=\frac{\mathrm{\Delta }𝑝}{\mathrm{\Delta }𝑡}=\frac{-60\text{ kg m/s}}{3\text{ s}}=-20\text{ N}$ The force is $20\text{ N}$.

Problem 6

Problem: A 60 N force is applied to push a box 5 meters across a floor. Calculate the work done. Solution: $𝑊=𝐹𝑑=60\text{ N}\times 5\text{ m}=300\text{ J}$ The work done is $300\text{ J}$.

Problem 7

Problem: A machine lifts a 600 kg object vertically at a constant speed of 3 m/s. Calculate the power developed. Solution: $𝐹=𝑚𝑔=600\text{ kg}\times 9.8{\text{ m/s}}^2=5880\text{ N}$ $𝑃=𝐹𝑣=5880\text{ N}\times 3\text{ m/s}=17640\text{ W}$ The power developed is $17640\text{ W}$.

Problem 8

Problem: Calculate the gravitational force between two 80 kg masses separated by 3 meters. Solution: $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}=6.674\times 10^{-11}{\text{ N m}}^2\mathrm{/}{\text{kg}}^2\times \frac{80\text{ kg}\times 80\text{ kg}}{(3\text{ m}{)}^2}=3.56\times 10^{-8}\text{ N}$ The gravitational force is $3.56\times 10^{-8}\text{ N}$.

Problem 9

Problem: A 1.5 kg ball is hit by a bat, changing its velocity from 6 m/s to 15 m/s in 0.3 seconds. Calculate the impulse imparted to the ball. Solution: $𝐽=\mathrm{\Delta }𝑝=𝑚({𝑣}_{𝑓}-{𝑣}_{𝑖})=1.5\text{ kg}\times (15\text{ m/s}-6\text{ m/s})=13.5\text{ kg m/s}$ The impulse is $13.5\text{ kg m/s}$.

Problem 10

Problem: Two ice skaters, one with mass 55 kg and the other with mass 70 kg, push off each other. If the 55 kg skater moves at 2.5 m/s, calculate the velocity of the 70 kg skater. Solution: ${𝑚}_1{𝑣}_1+{𝑚}_2{𝑣}_2=0$ $55\text{ kg}\times 2.5\text{ m/s}+70\text{ kg}\times {𝑣}_2=0$ $137.5\text{ kg m/s}+70\text{ kg}\times {𝑣}_2=0$ ${𝑣}_2=-\frac{137.5\text{ kg m/s}}{70\text{ kg}}=-1.96\text{ m/s}$ The velocity of the 70 kg skater is $1.96\text{ m/s}$ in the opposite direction.

Problem 11

Problem: A 10 kg block slides down a frictionless inclined plane making an angle of 30° with the horizontal. Calculate the component of the gravitational force acting down the plane. Solution: $𝐹=𝑚𝑔\sin 𝜃$ $𝐹=10\text{ kg}\times 9.8{\text{ m/s}}^2\times \sin 30^{\circ }=10\times 9.8\times 0.5=49\text{ N}$ The force is $49\text{ N}$.

Problem 12

Problem: A skydiver of mass 80 kg reaches terminal velocity where the drag force equals the gravitational force. Calculate the drag force. Solution: $𝐹=𝑚𝑔=80\text{ kg}\times 9.8{\text{ m/s}}^2=784\text{ N}$ The drag force is $784\text{ N}$.

Problem 13: Spring Force

Problem: A 2 kg mass hangs from a vertical spring, stretching it by 0.05 m. Calculate the spring constant $𝑘$. Solution: $𝐹=𝑘𝑥$ $𝑚𝑔=𝑘𝑥$ $𝑘=\frac{𝑚𝑔}{𝑥}=\frac{2\text{ kg}\times 9.8{\text{ m/s}}^2}{0.05\text{ m}}=392\text{ N/m}$ The spring constant is $392\text{ N/m}$.

Problem 14

Problem: A force of 20 N acts on an object moving it 10 m in the direction of the force. Calculate the work done. Solution: $𝑊=𝐹𝑑=20\text{ N}\times 10\text{ m}=200\text{ J}$ The work done is $200\text{ J}$.

Problem 15

Problem: A force of 100 N is applied to a surface area of 0.5 m². Calculate the pressure exerted on the surface. Solution: $𝑃=\frac{𝐹}{𝐴}=\frac{100\text{ N}}{0.5{\text{ m}}^2}=200\text{ Pa}$ The pressure is $200\text{ Pa}$.

Problem 16

Problem: A 12 kg box is placed on a horizontal surface with a coefficient of static friction of 0.45. Calculate the maximum static friction force. Solution: ${𝐹}_{𝑁}=𝑚𝑔=12\text{ kg}\times 9.8{\text{ m/s}}^2=117.6\text{ N}$ ${𝐹}_{𝑠}={𝜇}_{𝑠}{𝐹}_{𝑁}=0.45\times 117.6\text{ N}=52.92\text{ N}$ The maximum static friction force is $52.92\text{ N}$.

Problem 17

Problem: A cube of side length 0.5 m is submerged in water. Calculate the buoyant force acting on the cube. (Density of water = 1000 kg/m³) Solution: $𝑉={𝑠}^3=(0.5\text{ m}{)}^3=0.125{\text{ m}}^3$ ${𝐹}_{𝑏}=𝜌𝑔𝑉=1000{\text{ kg/m}}^3\times 9.8{\text{ m/s}}^2\times 0.125{\text{ m}}^3=1225\text{ N}$ The buoyant force is $1225\text{ N}$.

Problem 18

Problem: A 50 N force is applied to accelerate a 5 kg mass. Calculate the acceleration. Solution: $𝐹=𝑚𝑎$ $𝑎=\frac{𝐹}{𝑚}=\frac{50\text{ N}}{5\text{ kg}}=10{\text{ m/s}}^2$ The acceleration is $10{\text{ m/s}}^2$.

Problem 19

Problem: Two charges of $5\times 10^{-6}$ C each are separated by 0.2 m. Calculate the electric force between them. (Coulomb's constant $𝑘=8.99\times 10^9{\text{ N m}}^2\mathrm{/}{\text{C}}^2$) Solution: $𝐹=𝑘\frac{{𝑞}_1{𝑞}_2}{{𝑟}^2}=8.99\times 10^9\times \frac{(5\times 10^{-6}{)}^2}{0.2^2}=5.62\text{ N}$ The electric force is $5.62\text{ N}$.

Problem 20

Problem: A 2 kg object moves in a circle of radius 1 meter with a period of 2 seconds. Calculate the centripetal force. Solution:

1. Calculate the velocity: $𝑣=\frac{2𝜋𝑟}{𝑇}=\frac{2𝜋\times 1\text{ m}}{2\text{ s}}=𝜋\text{ m/s}$
2. Calculate the centripetal force: ${𝐹}_{𝑐}=\frac{𝑚{𝑣}^2}{𝑟}=\frac{2\text{ kg}\times (𝜋\text{ m/s}{)}^2}{1\text{ m}}=2{𝜋}^2\text{ N}$ The centripetal force is $2{𝜋}^2\text{ N}$.