Unit 4: Newton’s Second Law of Motion

The force of friction is equal to the applied force.

Since the crate doesn’t slide, its acceleration is equal to zero. Hence, the net force on the crate is also equal to zero. As shown in Figure Q4.3, the force of friction has exactly the same magnitude (but opposite direction) as the force applied on the crate.

It does not vary with speed.

At the microscopic level, frictional forces between two dry surfaces sliding and rubbing against each other are quite complicated. However, for most practical purposes, the force of friction can be considered independent of the speed and the area of contact between the two surfaces. So varying the speed does not change the force of friction on a sliding object.

Mass is more fundamental and is independent of location.

Mass is an intrinsic property and represents the quantity of matter in an object. A 10-kg rock has the same mass everywhere—on Earth, on the Moon, or in deep space far from any planet or star. Also, note that mass ($m$) is a scalar quantity.

Weight, however, is a vector quantity and represents the force of gravity on an object. Numerically, the magnitude of an object’s weight ($mg$) is equal to the product of its mass and the acceleration due to gravity at the location of the object. Since the gravitational force deceases with altitude, the same object weighs less at the top of a mountain than near sea level.

The mass of a spacesuit (without the astronaut in it) is more than 120 kg, which corresponds to a weight of 1200 N on Earth. On the surface of the Moon, however, the spacesuit s weight reduces to about 200 N. Its mass, of course, remains the same.

10 N

The weight of a 1-kilogram brick is equal to the force on it due to gravity, or \begin{align} \text{Weight} &= m g \nonumber\\[6pt] &= (1\,\text{kg})\, (10\,\text{m}/\text{s}^2) \nonumber\\[6pt] &= 10\,\text{N} \end{align}

The acceleration doubles.

Newton’s second law of motion can be expressed in the form \begin{equation} a = \left(\frac{1}{m}\right) F \end{equation} Based on this equation, the acceleration ($a$) of a sliding block is proportional to the net applied force ($F$). In other words, if the net force applied on the block is doubled, its acceleration is also doubled. Similarly, if $F$ is tripled, $a$ increases to three times the initial value. Note that the mass of the block ($m$) is constant in this situation.

Zero.

An object reaches terminal velocity when it stops accelerating and continues to fall down at a constant speed. In such a state, the upward force due to air drag balances the downward force due to gravity (or the object’s weight). As a result, the net force on the falling object is equal to zero.

The faster object.

Air resistance depends on the speed, size, and shape of an object. The resulting drag force increases with the object’s size and speed. If two objects have the same size (and shape), the faster object encounters greater air resistance.

8 N, downward

As shown in Figure Q4.41, two forces act on the falling ball: the force of gravity ($mg$) directed downward and the force of air resistance ($R$), which acts upward, opposite to the direction of motion. So you have \begin{align} mg &= (1\,\text{kg})\, (10\,\text{m}/\text{s}^2) \nonumber\\[6pt] &= 10\,\text{N}\quad (\text{downward}) \end{align} and \begin{equation} R = 2\,\text{N}\quad (\text{upward}) \end{equation} The net force on the ball is then equal to the difference between its weight and the force of air resistance: \begin{align} F_\text{net} &= mg - R \nonumber\\[6pt] &= 10\,\text{N} - 2\,\text{N} \nonumber\\[6pt] &= 8\,\text{N}\quad (\text{downward}) \end{align}

By applying Newton’s second law of motion, you can calculate the acceleration of the block as follows: \begin{align} a &= \frac{F_\text{net}}{m} \nonumber\\[6pt] &= \frac{200\,\text{N}}{40\,\text{kg}} \nonumber\\[6pt] &= 5\,\text{m}/\text{s}^2 \end{align}

No.

Inertia is a synonym for mass, which is a measure of the amount of material in an object. Relocating an object does not affect its composition—it will have the same mass anywhere. Thus, an object’s inertia on the Moon is the same as on Earth.

Observe how the boxes respond to an applied force.

Push the two boxes simultaneously with equal forces, using your left hand on one and your right on the other. The lighter box (filled with feathers) will have greater acceleration and move faster than the heavier box (filled with sand).

I would say, “You mean that no net force acts on the car!”

A car parked on a steep road, for example, experiences multiple forces, as shown in Figure Q4.85. First, there is the force of gravity (or weight), which acts in the downward direction. The second force on the car is called the normal (or support) force, and it is exerted perpendicular to the road’s surface. The car is prevented from sliding down the road by the force of friction, which is directed up the slope. These three forces, however, add up to zero. Your friend’s statement can be corrected by saying, “As long as a car is at rest, it experiences zero net force.”

Both forces are equal in strength.

Constant velocity means zero acceleration. According to Newton’s second law, the net force on the raindrop when it reaches terminal velocity must be equal to zero. Thus, the downward force of gravity on the raindrop is equal in magnitude to the upward force of air drag.

Upward.

When the parachute opens, the upward force of air resistance on the parachutist sharply increases and becomes much greater than her weight (downward force of gravity). As a result, the parachutist experiences a net force in the upward direction and starts to slow down—or, in other words, accelerate upward. As the falling speed decreases, air drag also becomes smaller. A new terminal speed is achieved when the force of air resistance becomes equal to the parachutist’s weight.

The more massive ball.

Assume Galileo dropped two solid balls of equal diameters, one made of steel and the other of wood. In this case, the downward force of gravity ($m_\text{s}\,g$) on the steel ball is much larger than the force of gravity ($m_\text{w}\,g$) on the lighter wooden ball.

Since the two balls have the same shape and size, the upward force ($R$) of air drag will depend on speed only. So, initially, the steel ball will experience greater downward acceleration ($g - R/m_\text{s}$) compared with the acceleration ($g - R/m_\text{w}$) of the wooden ball. Also, the steel ball will have to gain a much higher speed than the wooden ball before it reaches terminal velocity. So, overall, the heavier steel ball moves faster and strikes the ground first.

No.

Velocity is a vector quantity, with magnitude and direction, and any change in the speed or direction of motion of an object requires acceleration. While moving on a curved path, an object continuously changes its direction of motion, which means it continuously accelerates. According to Newton’s second law, acceleration is not possible without an applied force.

10 N in the rope between the left block and the center block, and 20 N in the rope between the center block and the right block.

Since the three blocks (of mass $m$ each) are connected, they all move together with the same acceleration ($a$). According to Newton’s second law, the force pulling the first block from the left is given by $F = m a$. Also, you will need twice the force ($2F$) to accelerate two blocks and a force equal to $3F$ for all three blocks to have an acceleration $a$.

In Figure Q4.110(b), you see that the tension $T_1$ in the first rope accelerates a single block, while the tension $T_2$ in the second rope accelerates two blocks. Therefore, $T_2 = 2F$ has twice the value of $T_1 = F$. Similarly, the tension in the rope held by the hand is given as \begin{equation} T_3 = 3 F = 30\,\text{N} \end{equation} So, you have \begin{equation} F = \frac{30\,\text{N}}{3} = 10\,\text{N} \end{equation} and you can write \begin{equation} T_1 = F = 10\,\text{N} \end{equation} and \begin{equation} T_2 = 2F = 20\,\text{N} \end{equation}

The flat sheet of paper falls more slowly.

The masses and, consequently, the weights of the two sheets are equal. The flat sheet of paper, however, has greater surface area than the wadded one. As a result, when both have the same speed, the upward force of air resistance (or drag) on the flat sheet is greater than on the wadded one. Consequently, the flat sheet reaches terminal velocity more quickly and falls slower than the wadded one.

The air resistance is the same in both situations.

The sheet of paper has the same weight whether it is flat or crumpled into a ball. However, when it is flat, the sheet of paper balances the forces (i.e., air drag equals the weight) at a smaller terminal speed than when the sheet is wadded into a ball. While falling at its terminal speed, an object always experiences zero net force, where the upward force of air resistance is equal to the downward force of gravity (or weight).