Force: Mathematical problem & solution-3

Problem 1: Force and Acceleration

Problem: A 5 kg object is subjected to a force that accelerates it at 3 m/s². Calculate the force applied.

Solution: Using Newton's second law, $𝐹=𝑚𝑎$:

$$𝐹=5\text{ kg}\times 3{\text{ m/s}}^2=15\text{ N}$$

The force applied is $15\text{ N}$.

Problem 2: Force and Friction

Problem: A 10 kg box is being pushed across a horizontal surface with a coefficient of kinetic friction of 0.4. Calculate the force required to keep the box moving at a constant velocity.

Solution:

1. Calculate the normal force (${𝐹}_{𝑁}$):

$${𝐹}_{𝑁}=𝑚𝑔=10\text{ kg}\times 9.8{\text{ m/s}}^2=98\text{ N}$$

2. Calculate the frictional force (${𝐹}_{𝑓}$):

$${𝐹}_{𝑓}=𝜇{𝐹}_{𝑁}=0.4\times 98\text{ N}=39.2\text{ N}$$

The force required is $39.2\text{ N}$.

Problem 3: Force in Circular Motion

Problem: A 2 kg object is whirled in a horizontal circle of radius 1 meter at a speed of 4 m/s. Calculate the centripetal force.

Solution: Using ${𝐹}_{𝑐}=\frac{𝑚{𝑣}^2}{𝑟}$:

$${𝐹}_{𝑐}=\frac{2\text{ kg}\times (4\text{ m/s}{)}^2}{1\text{ m}}=32\text{ N}$$

The centripetal force is $32\text{ N}$.

Problem 4: Hooke's Law

Problem: A spring with a spring constant of 200 N/m is compressed by 0.05 meters. Calculate the force exerted by the spring.

Solution: Using Hooke's law, $𝐹=-𝑘𝑥$:

$$𝐹=-200\text{ N/m}\times 0.05\text{ m}=-10\text{ N}$$

The force exerted is $10\text{ N}$.

Problem 5: Momentum Change

Problem: A 3 kg ball moving at 10 m/s is brought to rest in 2 seconds by a constant force. Calculate the force.

Solution:

1. Calculate the change in momentum ($\mathrm{\Delta }𝑝$):

$$\mathrm{\Delta }𝑝=𝑚({𝑣}_{𝑓}-{𝑣}_{𝑖})=3\text{ kg}\times (0-10\text{ m/s})=-30\text{ kg m/s}$$

2. Calculate the force ($𝐹$):

$$𝐹=\frac{\mathrm{\Delta }𝑝}{\mathrm{\Delta }𝑡}=\frac{-30\text{ kg m/s}}{2\text{ s}}=-15\text{ N}$$

The force is $15\text{ N}$.

Problem 6: Work Done by Force

Problem: A 50 N force is applied to push a box 4 meters across a floor. Calculate the work done.

Solution: Using $𝑊=𝐹𝑑$:

$$𝑊=50\text{ N}\times 4\text{ m}=200\text{ J}$$

The work done is $200\text{ J}$.

Problem 7: Power Developed

Problem: A machine lifts a 500 kg object vertically at a constant speed of 2 m/s. Calculate the power developed.

Solution:

1. Calculate the force ($𝐹$):

$$𝐹=𝑚𝑔=500\text{ kg}\times 9.8{\text{ m/s}}^2=4900\text{ N}$$

2. Calculate the power ($𝑃$):

$$𝑃=𝐹𝑣=4900\text{ N}\times 2\text{ m/s}=9800\text{ W}$$

The power developed is $9800\text{ W}$.

Problem 8: Gravitational Force

Problem: Calculate the gravitational force between two 100 kg masses separated by 2 meters.

Solution: Using Newton's law of gravitation, $𝐹=𝐺\frac{{𝑚}_1{𝑚}_2}{{𝑟}^2}$, where $𝐺=6.674\times 10^{-11}{\text{ N m}}^2\mathrm{/}{\text{kg}}^2$:

$$𝐹=6.674\times 10^{-11}{\text{ N m}}^2\mathrm{/}{\text{kg}}^2\times \frac{100\text{ kg}\times 100\text{ kg}}{(2\text{ m}{)}^2}=1.6685\times 10^{-8}\text{ N}$$

The gravitational force is $1.6685\times 10^{-8}\text{ N}$.

Problem 9: Impulse

Problem: A 2 kg ball is hit by a bat, changing its velocity from 5 m/s to 10 m/s in 0.2 seconds. Calculate the impulse imparted to the ball.

Solution: Impulse ($𝐽$) is given by the change in momentum:

$$𝐽=\mathrm{\Delta }𝑝=𝑚({𝑣}_{𝑓}-{𝑣}_{𝑖})=2\text{ kg}\times (10\text{ m/s}-5\text{ m/s})=10\text{ kg m/s}$$

The impulse is $10\text{ kg m/s}$.

Problem 10: Newton's Third Law

Problem: Two ice skaters, one with mass 70 kg and the other with mass 50 kg, push off each other. If the 70 kg skater moves at 2 m/s, calculate the velocity of the 50 kg skater.

Solution: Using conservation of momentum:

$${𝑚}_1{𝑣}_1+{𝑚}_2{𝑣}_2=0$$

Given:

• ${𝑚}_1=70\text{ kg}$
• ${𝑣}_1=2\text{ m/s}$
• ${𝑚}_2=50\text{ kg}$
• ${𝑣}_2=?$

$$70\text{ kg}\times 2\text{ m/s}+50\text{ kg}\times {𝑣}_2=0$$

$$140\text{ kg m/s}+50\text{ kg}\times {𝑣}_2=0$$

$${𝑣}_2=-\frac{140\text{ kg m/s}}{50\text{ kg}}=-2.8\text{ m/s}$$

The velocity of the 50 kg skater is $2.8\text{ m/s}$ in the opposite direction.

These problems cover a range of scenarios involving forces, providing a comprehensive review of how force is applied in different contexts.