Exercises: The laws of dynamics

A high-speed train travels along a straight section of track at a constant speed of $300\ \text{km/h}$.

1. a) What is the acceleration of the train on that section?
2. b) Applying Newton's first law, what is the net force acting on the train?
3. c) If the thrust of the engines is $5 \times 10^{4}\ \text{N}$, what is the total friction force opposing forward motion? Justify your answer.

In the lab, a net force of $24\ \text{N}$ is applied to a dynamics cart of $3\ \text{kg}$ moving along a horizontal track.

1. a) Write the expression for Newton's second law, $\sum \vec{F} = m \cdot \vec{a}$, and make acceleration the subject.
2. b) Calculate the value of the acceleration of the cart.

A worker leaves a container of $10\ \text{kg}$ on a loading ramp that forms $30^\circ$ with the horizontal (take $g = 10\ \text{m/s}^2$, $\sen 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.87$). The diagram shows the container on the ramp.

1. a) Calculate the weight of the container.
2. b) Calculate the component of the weight parallel to the ramp, $P_x = P \times \sen 30^\circ$.
3. c) Calculate the perpendicular component, $P_y = P \times \cos 30^\circ$, and state which force from the plane balances it.

A crate of $5\ \text{kg}$ rests on the horizontal floor of a workshop. The coefficient of friction between the crate and the floor is $\mu = 0.4$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the normal force that the floor exerts on the crate.
2. b) Calculate the friction force when the crate slides, $F_r = \mu \times N$.
3. c) If the crate is pushed with a horizontal force of $20\ \text{N}$, will it move at a constant speed? Give reasons for your answer.

A bat hits a baseball, exerting a forward force of $200\ \text{N}$ on it. The diagram shows the instant of impact.

1. a) State Newton's third law (action and reaction).
2. b) What force does the ball exert on the bat? State its magnitude, direction, and sense.
3. c) Explain why, despite both forces having the same magnitude, the ball flies off while the bat barely changes its motion.

A lamp of $4\ \text{kg}$ hangs at rest from a cable attached to the ceiling (take $g = 10\ \text{m/s}^2$). The diagram shows the lamp hanging from the cable.

1. a) Draw in your notebook the forces acting on the lamp and state which body exerts each one.
2. b) Calculate the tension in the cable.
3. c) If the cable can only withstand a maximum tension of $100\ \text{N}$, what is the maximum mass of the lamp that can be hung?

You can do it on paper and send a photo with your answer.

The motor of an electric scooter exerts a net force of $40\ \text{N}$. The diagram compares the two situations.

1. a) Calculate the acceleration when only the scooter travels, having a mass of $20\ \text{kg}$.
2. b) Calculate the acceleration when it carries a backpack and the total mass is $40\ \text{kg}$.
3. c) Explain, using $\vec{a} = \frac{\vec{F}}{m}$, why doubling the mass halves the acceleration.

Two horizontal forces along the same line act on a crate of $4\ \text{kg}$ on the loading dock: $30\ \text{N}$ to the right and $18\ \text{N}$ to the left. The diagram shows the crate with both forces.

1. a) Calculate the net force on the crate, stating its direction.
2. b) Calculate the acceleration of the crate.
3. c) What additional force would need to be applied for the crate to move at a constant speed? Justify your answer using Newton's first law.

A worker pushes a workshop cart of $8\ \text{kg}$ with a horizontal force of $52\ \text{N}$. The coefficient of friction with the floor is $\mu = 0.5$ (take $g = 10\ \text{m/s}^2$). The diagram shows the forces acting on the cart.

1. a) Calculate the normal force and the friction force.
2. b) Calculate the net horizontal force on the cart.
3. c) Calculate its acceleration.

In a lab experiment, a cart of $2\ \text{kg}$ is released down an inclined plane of $30^\circ$, with negligible friction (take $g = 10\ \text{m/s}^2$, $\sen 30^\circ = 0.5$ and $\cos 30^\circ \approx 0.87$).

1. a) Calculate the components of the weight parallel and perpendicular to the plane.
2. b) Calculate the acceleration of the cart.
3. c) If it starts from rest, calculate the time it takes to travel $10\ \text{m}$ along the plane and its final velocity (use $x = \frac{1}{2} \times a \times t^2$ and $v = a \times t$).

A rescue sled of $40\ \text{kg}$ goes down a snowy slope of $37^\circ$ with a coefficient of friction of $\mu = 0.5$ (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$). The diagram shows the sled on the slope with the forces acting on it.

1. a) Calculate the component of the weight parallel to the slope.
2. b) Calculate the normal force and the friction force.
3. c) Calculate the acceleration of the sled and verify that it does not depend on its mass: would a loaded sled go down with the same acceleration as an empty one? Give reasons why.

To pull a garden cart of $10\ \text{kg}$, a rope is pulled with a force of $100\ \text{N}$ at an angle of $37^\circ$ to the horizontal. The coefficient of friction with the ground is $\mu = 0.25$ (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$). The diagram shows the cart and the rope.

1. a) Resolve the force of the rope into its horizontal and vertical components.
2. b) Taking into account that the rope "lifts" the cart slightly, calculate the normal force and the friction force.
3. c) Calculate the net horizontal force and the acceleration of the cart.

Two masses, $m_1 = 3\ \text{kg}$ and $m_2 = 5\ \text{kg}$, hang from an ideal pulley in the lab, connected by an inextensible string (take $g = 10\ \text{m/s}^2$). As the string is inextensible, both masses move with the same acceleration $a$, and the tension $T$ is the same throughout the string. The diagram shows the pulley setup.

1. a) Draw in your notebook the forces acting on each mass.
2. b) Write Newton's second law for each mass: $T - m_1 \times g = m_1 \times a$ for the one going up and $m_2 \times g - T = m_2 \times a$ for the one going down.
3. c) Add the two equations to eliminate $T$ and calculate the acceleration of the system.

You can do it on paper and send a photo with your answer.

A toy locomotive of $2\ \text{kg}$ pulls two cars, of $2\ \text{kg}$ and $4\ \text{kg}$, connected by couplings, exerting a tractive force of $24\ \text{N}$ (consider friction negligible). The train consists of, from front to back: locomotive, $2\ \text{kg}$ car, and $4\ \text{kg}$ car.

1. a) Calculate the acceleration of the system.
2. b) Applying Newton's second law only to the $4\ \text{kg}$ car, which is at the back, calculate the tension in the coupling between the two cars.
3. c) Calculate the tension in the coupling between the locomotive and the first car and explain why it is greater than the one in the previous part.

A person of $60\ \text{kg}$ weighs themselves on a scale inside an elevator (take $g = 10\ \text{m/s}^2$). The diagram shows the person on the scale.

1. a) What does the scale read with the elevator at rest?
2. b) The elevator starts moving upward with an acceleration of $1.5\ \text{m/s}^2$. Apply Newton's second law vertically, $N - P = m \times a$, and calculate the reading on the scale.
3. c) Has the person's real weight changed? Explain which quantity has changed.

To move a moving crate of $20\ \text{kg}$ along the floor, the coefficient of static friction is $\mu_e = 0.5$ and the coefficient of kinetic friction is $\mu_d = 0.4$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the minimum horizontal force required to set the crate in motion.
2. b) Calculate the horizontal force required to keep it moving at a constant speed.
3. c) In light of the results, explain why it takes more effort to start moving the crate than to keep it moving.

A package of $4\ \text{kg}$ slides from the top of a warehouse ramp that forms $37^\circ$ with the horizontal. The coefficient of friction is $\mu = 0.25$ (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$).

1. a) Draw in your notebook the forces acting on the package and calculate the magnitude of each.
2. b) Calculate the acceleration with which it moves down.
3. c) If the ramp is $8\ \text{m}$ long and the package starts from rest, calculate the speed with which it reaches the bottom and the time it takes (use $v^2 = 2 \times a \times x$ and $v = a \times t$).

You can do it on paper and send a photo with your answer.

A block of $6\ \text{kg}$ is on a horizontal table, connected by a string passing over an ideal pulley to another hanging block of $4\ \text{kg}$. The coefficient of friction between the first block and the table is $\mu = 0.25$ (take $g = 10\ \text{m/s}^2$). The diagram shows the setup.

1. a) Draw in your notebook the forces acting on each block.
2. b) Calculate the friction force on the block on the table.
3. c) Set up Newton's second law for each block, taking into account that both have the same acceleration $a$ and that the tension $T$ is the same throughout the string, and calculate the value of $a$.
4. d) Calculate the tension in the string.

You can do it on paper and send a photo with your answer.

A remote-controlled car of $2\ \text{kg}$ moves in a straight line. Its velocity changes as follows: from $t = 0$ to $t = 4\ \text{s}$ it goes from $0$ to $8\ \text{m/s}$ uniformly; from $t = 4\ \text{s}$ to $t = 10\ \text{s}$ it maintains a velocity of $8\ \text{m/s}$; from $t = 10\ \text{s}$ to $t = 14\ \text{s}$ it brakes uniformly until it stops.

1. a) Draw the velocity-time graph in your notebook.
2. b) Calculate the acceleration of each of the three sections.
3. c) Applying Newton's second law, calculate the net force on the car in each section and state its direction.
4. d) In the second section the net force is zero: explain whether this means that the engine exerts no force.

You can do it on paper and send a photo with your answer.

To determine the coefficient of friction between a board and a block, the board is tilted gradually and it is observed that, when it forms $37^\circ$ with the horizontal, the block moves down at a constant speed (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$).

1. a) Draw in your notebook the forces on the block and explain why the net force is zero.
2. b) Write the equations of equilibrium parallel to the plane and perpendicular to it.
3. c) Deduce that under these conditions $\mu = \frac{\sen \alpha}{\cos \alpha} = \tan \alpha$ and calculate the value of $\mu$.
4. d) Predict, showing the relevant calculation, the acceleration the block would have on that same ramp if the coefficient of friction were $\mu = 0.5$.

You can do it on paper and send a photo with your answer.

A cart of $2\ \text{kg}$ is launched upward along an inclined plane of $37^\circ$ with an initial velocity of $8\ \text{m/s}$. The coefficient of kinetic friction is $\mu_d = 0.25$ and the static coefficient is $\mu_e = 0.5$ (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$). The diagram shows the cart at the instant of launching.

1. a) Draw in your notebook the forces on the cart while it moves up (note: friction opposes the motion) and calculate its acceleration.
2. b) Calculate the distance it travels until it stops and the time it takes (use $v^2 = v_0^2 - 2 \times a \times x$ and $v = v_0 - a \times t$).
3. c) Once stopped at the top, does the cart slide back down? Compare the component of the weight parallel to the plane with the maximum static friction force and give reasons for your answer.

You can do it on paper and send a photo with your answer.

In an Atwood machine in the lab, masses $m_1 = 2\ \text{kg}$ and $m_2 = 6\ \text{kg}$ hang from an inextensible string passing over an ideal pulley (take $g = 10\ \text{m/s}^2$).

1. a) Draw in your notebook the forces acting on each mass.
2. b) Set up the system of two equations with the unknowns $a$ and $T$, and solve it: calculate the acceleration of the system and the tension in the string.
3. c) Verify that the tension is greater than the weight of $m_1$ and less than the weight of $m_2$, and explain why this must be the case while the system is accelerating.

You can do it on paper and send a photo with your answer.

A student calculates the acceleration of a block of $3\ \text{kg}$ sliding down a frictionless inclined plane of $30^\circ$ and writes: "since the only force is the weight, $a = \frac{F}{m} = \frac{m \times g}{m} = g = 10\ \text{m/s}^2$" (take $g = 10\ \text{m/s}^2$ and $\sen 30^\circ = 0.5$).

1. a) Identify the two errors in the reasoning: which force has been omitted and which component of the weight actually accelerates the block.
2. b) Calculate the correct acceleration of the block.
3. c) Calculate the actual net force on the block and compare it with its weight.

A container of $5\ \text{kg}$ is on a ramp of $37^\circ$ in a cargo port, connected by a cable passing over an ideal pulley to a counterweight of $5\ \text{kg}$ hanging vertically. The coefficient of friction between the container and the ramp is $\mu = 0.2$ (take $g = 10\ \text{m/s}^2$, $\sen 37^\circ \approx 0.6$ and $\cos 37^\circ \approx 0.8$). The diagram shows the ramp setup with the counterweight.

1. a) Draw in your notebook the forces acting on the container and on the counterweight.
2. b) Calculate the component of the container's weight parallel to the ramp and the friction force, and reason in which direction the system moves.
3. c) Calculate the acceleration of the system.
4. d) Calculate the tension in the cable.

You can do it on paper and send a photo with your answer.