Each of the following (2-dimensional) free-body diagrams shows the directions of all the forces that are acting on an object. If the magnitude of each force can be adjusted to any non-zero value but its direction cannot be changed, which of the objects can be put into equilibrium?
Figure3.19.1.
Answer.
(1), (2), and (5)
Calculation3.19.2.Teapot.
A person pushes a cart to the left at constant velocity across a level floor. A teapot sits on the cart without slipping. Assume there is no air resistance. Choose the correct free-body diagram for the teapot.
Figure3.19.2.
Answer.
(1)
Calculation3.19.3.Elevator.
The force of gravity on an 80-kg person is about \(800 \mathrm{~N}\text{.}\) Suppose this person is standing on a scale in an elevator that is moving upwards but slowing down with an acceleration magnitude of \(1 \mathrm{~m/s^2}\text{.}\) What value does the scale read?
Answer.
720 N
Calculation3.19.4.Pulling on a rope.
An 80-kg person stands at rest on a scale while pulling vertically downward on a rope that is hanging at rest directly above them. Use \(g = 9.80 \mathrm{~m/s^2}\text{.}\) With what force magnitude must the rope’s tension be pulling on the person so that the scale reads \(500 \mathrm{~N}\text{?}\) With what force magnitude must the rope’s tension be pulling on the person so that the scale reads 25% of the person’s weight? What is the critical magnitude of tension in the rope so that the person just begins to lift off the scale?
Answer1.
\(284 \mathrm{~N}\)
Answer2.
\(588 \mathrm{~N}\)
Answer3.
\(784 \mathrm{~N}\)
Calculation3.19.5.Dropping a ball.
On a windy day, a 1.50-kg ball is dropped from rest from a height of 19.6 meters above the Earth’s surface. A steady wind pushes on the falling ball with a constant, horizontal force of 8.40 N, to the right. What is the total displacement of the ball from its initial location to its point of impact on the ground? Use a standard coordinate system with the origin located at the ball’s initial location. Also, use \(g = 9.80 \mathrm{~m/s^2}\text{.}\) Assume no air effects other than the steady wind.
In which of the following situations is the friction force that is acting on the object not in the opposite direction of the object’s velocity? Choose all that apply.
A block slides to rest on a stationary table.
A block, initially at rest in the bed of a stationary pick-up truck, begins to slide to the back of the truck as the truck accelerates and moves forwards.
A block initially at rest in the bed of a stationary pick-up truck does not slide but begins to move forwards with the truck as the truck accelerates and moves forwards.
A car initially at rest does a burnout. (It begins to accelerate and move forward as the wheels spin on the ground.)
Answer.
(B), (C), (D)
Calculation3.19.7.Pushing on a block.
A \(1.40 \mathrm{~kg}\) block is at rest on a level table. The coefficient of static friction between the table and the block is 0.40, and the coefficient of kinetic friction between the two is 0.10. A person then applies a 6.00 N force at an angle of 30.0\(\deg\) downward relative to the positive-x direction. Use \(g = 9.80 \mathrm{~m/s^2}\) and a standard coordinate system. What is the magnitude of the friction force acting between the block and table? What is the acceleration of the block? What is the acceleration of the block if the person applies the \(6.00 \mathrm{~N}\) force at an angle of 30.0\(\deg\) upward relative to the positive-x direction?
Answer1.
\(5.20 \mathrm{~N}\)
Answer2.
\(\vec{a} = 0 \frac{m}{s^2}\)
Answer3.
\(\vec{a} = 2.95 \frac{m}{s^2} \hat{x}\)
Calculation3.19.8.Angled ramp with a pulley.
A box (\(m_1 = 10.5 \mathrm{~kg}\)) is on an inclined plane with negligible friction between the box and the incline. This box is attached via an ideal string and pulley to the hanging mass (\(m_1 = 5.5 \mathrm{~kg}\)), as shown. The incline make an angle of 20.0° with respect to the horizontal. Use \(g = 9.80 \mathrm{~m/s^2}\text{.}\) Rank the magnitude of accelerations for \(m_1\) and \(m_2\text{.}\) Do we necessarily know the direction of acceleration of \(m_1\) without doing any calculations? Find the magnitude of acceleration for \(m_1\text{.}\) What direction is the acceleration of \(m_1\text{?}\)
Figure3.19.3.
Answer1.
\(|\vec{a}_1| = |\vec{a}_2|\)
Answer2.
No, but we can just guess a direction. For example, if we assume \(m_1\) accelerates up the ramp but our analysis results in a negative acceleration for \(m_1\text{,}\) that means we picked the wrong direction, but the magnitude of our result is correct.
Answer3.
\(1.17 \mathrm{~m/s^2}\)
Answer4.
Up the incline.
SubsectionStudy
Explanation3.19.9.Skydiver.
When a skydiver jumps from a plane, they quickly become subject to a substantial force known as air resistance or air drag. This force becomes larger when the skydiver’s velocity becomes larger.
During the first part of falling, the skydiver is accelerating downward. Draw and label a freebody diagram for the skydiver. Rank the magnitudes of all forces on the skydiver.
(a)Constant Velocity.
Eventually, the skydiver stops accelerating and moves with constant velocity (this is known as terminal velocity). Draw and label a free-body diagram for the skydiver. Rank the magnitudes of all forces on the skydiver.
(b)Throwing an Object.
Suppose the skydiver throws an object downward so that the object’s speed is greater than the terminal velocity, with respect to the ground. In this situation, the acceleration of the object will be upward. Draw and label a free-body diagram for the object. Rank the magnitudes of all forces on the object.
Explanation3.19.10.Water Slide.
You slide down a water slide of length \(L\) that makes an angle \(\theta\) with the horizontal, starting from rest. Do not neglect friction (use \(\mu_k\) as the coefficient of kinetic friction). Determine the amount of time it takes you to reach the bottom of the slide.
Tip.
To make sense of your answer, use special-case analysis in the case that friction can be neglected.
Explanation3.19.11.Walking a Cat.
You are trying to exercise your cat by walking them around the room with a harness - a jacket is strapped around the cat with a leash that you can pull. To get your cat moving, you pull on the leash at an angle \(\theta\) above the horizontal (note this is not how you should train your cat with a harness). Annoyed, your cat flops onto their side where there is a sizable friction between the cat and the floor. As you pull your cat, you notice they move with a constant speed.
From previous experiments with this harness, you know that your "pull" at the given angle has a vertical component equal to half the weight of your cat.
(a)Rank the forces.
During the pull, rank the magnitude of the forces acting on the cat.
To assist you, you add an additional rope hanging from the ceiling that always stays vertical but the attachment is free to move along the ceiling (like something you would find in an action movie). This new rope’s tension has a magnitude equal to the vertical component of your pull. You now pull on the leash with the same magnitude and direction as before.
(b)Explain.
Now, you notice the cat speeds up. Explain why this occurs.
Explanation3.19.12.Supported Block.
The two blocks in the figure are being pushed so that they move together with increasing speed and that block B does not move vertically.
Figure3.19.4.Two blocks moving together.
Sketch a free-body diagram for each block and explicitly rank the magnitudes of the forces acting on each block.
A*R*C*S3.19.13.Stacked Blocks.
The blocks below have different masses, \(m_1\) and \(m_2\text{.}\) Each block is connected to a horizontal rope; the upper rope is connected to the wall and the lower rope is being pulled to the right with known tension \(T\text{.}\) All coefficients of kinetic friction \(\mu_k\) between all surfaces are equal and nonzero. Determine the tension in the upper rope and the acceleration of the lower block?
Figure3.19.5.Two blocks attached to ropes.
Tip.
As part of your sensemaking, evaluate your answer in at least one special case, including a detailed explanation for what the answer should be for the case you chose.
SubsectionApply
Explanation3.19.14.Three Cars Redux.
Recall Exercise 3.4.4. Answer this question again. As part of your explanation, discuss how your answer, your reasoning, or your understanding has changed since you first thought about this task.
Explanation3.19.15.The Angled Block.
In the two situations depicted below, an angled block is at rest on a ramp. In each case, a hand pushes on the block with constant force. In Case A, the hand pushes on the upper horizontal surface of the block. In Case B, the hand pushes on the upper angled surface with a constant force of the same magnitude. In both cases, the block remains at rest on the ramp.
Figure3.19.6.A hand pushes on different surfaces on a block on a ramp.
Part A: Is the magnitude of the net force on the block in Case B greater than, less than, or equal to the magnitude of the net force in Case A?
Part B: Is the magnitude of the frictional force exerted on the block by the ramp in Case B greater than, less than, or equal to the magnitude of the frictional force in Case A?
Explanation3.19.16.Accelerating Truck.
A truck is traveling in a straight line on level ground, and is accelerating uniformly. Two ropes are tied to the back of the truck. The other end of each rope is tied to a bucket: the bucket tied to rope 1 has a larger mass than the bucket tied to rope 2. You notice that each rope is hanging from the back of the truck at a fixed (but potentially different) angle. Is the angle of rope 1 greater than, less than, or equal to the angle of rope 2?
Figure3.19.7.A bucket suspended from the back of an accelerating truck.
A*R*C*S3.19.17.Rope Tension.
An athlete of mass m is training by hanging from two ropes attached to the ceiling, as shown in the figure. The angles \(\theta\) and \(\alpha\) are different. Determine the magnitude of the tension in each rope.
Figure3.19.8.An athlete suspended by two ropes.
Tip.
As part of your symbolic sensemaking (part 3c), use special-case analysis in the case that the two angles are equal to each other.
Explanation3.19.18.Two Books in an Elevator.
The two books from class (shown below) are in an elevator that is moving downward. As the elevator approaches the ground floor, its speed decreases. Identify and rank all forces acting on the two books by magnitude, from largest to smallest.
Figure3.19.9.Two books in an elevator.
Tip.
Free-body diagrams can be very helpful! Your reasoning should specifically reference how you used Newton’s Laws.
A*R*C*S3.19.19.Tilted Ramp.
The two boxes below are initially at rest, connected by ropes and sitting on tables. Determine the acceleration of the system, assuming that the boxes begin to slide and that both surfaces have nonzero friction (\(\mu_k\)).
Figure3.19.10.Two blocks, connected by a rope on a tilted ramp.
Tip.
As part of your sensemaking, evaluate your answer in at least one special case, including a detailed explanation for what the answer should be for the case you chose.
A*R*C*S3.19.20.Moving Blocks.
Shown below are two blocks connected by an ideal string that passes over a massless, frictionless pulley. The mass of the larger block sits on a flat, frictionless table, and has four times the mass of the smaller block, which hangs vertically. Determine the speed of each block when the larger block reaches the edge of the table, a distance of 0.75 m.
Figure3.19.11.Two blocks, connected by a rope.
ReferencesReferences
[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
