4 hard (long) problems- problem set.

• ForKeli
In summary, the professor hit a lever, and the bucket fell into the well. The bucket was suspended from a rope of negligible mass which was wrapped around a cylinder of mass M and radius R. The cylinder was mounted on a frictionless horizontal axis over the well. The dynamical equations of motion were used to find an expression for the acceleration of the vucket as it fell down the well shaft. Three seconds after the bicket started its descend, the professor heard the splash of the well ater. If the mass of the bicket is 4 kg and the mass of the cylinder is 12 kg, find
ForKeli
1)A physics professor reaches for a well bucket of mass m, the professor hits a lever and the bucket falls into the well. The bucket is suspended from a rope of negligible mass which is wrapped around a sold cylinder of mass M and radius R. The cylinder is mounted on a frictionless horizontal axis over the well. (a) Use the dynamical equations of motion to find an expression for the acceleration of the vucket as it falls down the well shaft, (b) Three seconds after the bicket started its descend, the professor heard the splash of the well ater. If the mass of the bicket is 4 kg and the mass of the cylinder is 12 kg, find the depth of well. (Take the speed of the sound in air as 340 m/s. (c) What is the speed of the buckert just before it hits the water?

2) The cutting blade assembly on a radial-arm saw has a mass of 4 kg. It is pulled along a pair of frictionless horizontal rails aligned with the x-axis by a force F(x). The position of the blade assembly as a function of time is : x=(0.18 m/s^2)t^2 - (0.030 m/s^3)t^3. (a) Draw a free-body diagram for the blade assembly. Identify the source of each force in your diagram. (b) Find the net force acting on the blade assembly as a function of time. (c) Sketch a graph for velocity vs. time - for what values of time is the net force positive? Negative? Zero?

3) Two prolific painters concoct a hair-brain scheme for painting a spherical-shaped observatory dome. One painter has invented a pair of roller skates that will eject paint only when the wheels of the skates are in contact with a surface. The daredevil of the two, wearing the roller skates, starts with negligible velocity at the top of the dome and coasts down over the dome's surface. As long as the wheels are in contact, the surface of the dome will receive a coat of paint. The more sensible painter is positioned with a net some distance from the base of the observatory to break the daredevils fall. The mass of the painter, including the skates, is 80 kg and the radius of the dome is 8 m. A test run is made to demonstrate how much of the dome can be painted in one pass. Assume that the painters mass is constant over this one run. Neglect friction and air resistance. (a) Describe in angles (relative the vertical) the portion of the dome that the daredevil would paint and the portion that would remain unpainted. (b) If the net is removed, what velocity would the painter have when he impacts the ground? (c) What impulse would the ground deliver to the painter in order to stop him? Give the direction relative to the vertical. (d) A person can just survive a full body collision if the average force is less than 72,000 Newtons. If it takes the ground .05 seconds to stop the painter, is it likely he will survive the fall? Explain with physics principles. (e) Where should the assistant place the net from the dome to break the daredevils fall? Identify physics principle to use.

4) An accident occurred when the driver of an automobile traveling down an 8% grade slope hit a parked car. A photograph of the accident scene revealed skid marks leading directly to the parked car. Fortunately, no one was injured by this rear end collision. The police officers responding to the call took a statement from the driver, measured the length of the skid marks, and performed a number of skid tests at the accident site. In their final report, the officers reported that the skid marke were 30m long, that the coefficient of friction was .45, and that the driver recalled slamming on her breaks and sliding into the parked car. A footnote in the report indicates that the skid tests were performed over the first 6 m of the skid marks. Based on the evidence, did the driver exceed the 25mph speed limit or should she challenge the incriminating evidence?(a) Assume that the car is a point mass and draw a free-body diagram showing all the forces acting on it as it slides down the hill. (b) Find the expression for the acclereation of the car after the driver slammed on the breaks. (c) The police made the assumption that the coefficient of friction over the first 6m was the same as that of the rest of the skid marks. Using this assumption, calculate the acceleration of the car. How does it compare with the acceleration due to gravity? (d) Find an expression for the time it takes the car to stop after the driver depresses the brakes.(e) Assume an initial velocity of 25 mph, how far would the car have traveled before stopping? (f) If the case goes to court, the judge would want to know the minimum velocity of the car before the driver applied the breaks. Calculate the minimum initial velocity of the car. (g) Based on the evidence in the accident report and the assumptions used by the police, do you recommend the driver be charged with speeding?

did you try them?

yep. For the last one, I know that the forces acting on the car that caused the collision are : F(g), F(normal), and sliding force - I think I'm missing something. I used a=gf; f being the coefficient of friction calculated by the police, to find acceleration. Since the car is on a slope F(g) would be F(cos[theta]) for the y-direction and F(sin[theta]) for the x-direction.In the x-direction are the forces F(sin[theta]) + F(sliding friction)=ma. In the y-direction are the forces -F(cos[theta])+F(normal)=ma. After she slams the breaks its all sliding friction that's causing the car to travel. Since she was already in motion her initial velocity cannot be zero, but after hitting the car her final velocity would be. I could use the distance slid and the acceleration found to find the time using uniform acceleration equations and the initial velocity.
I think.
The others I haven't much of an idea since we just started those topics on circular motion and torque the day before this problem set was given. I am still trying to grasp those.

ForKeli said:
yep. For the last one, I know that the forces acting on the car that caused the collision are : F(g), F(normal), and sliding force - I think I'm missing something.
So far, that's correct.
I used a=gf; f being the coefficient of friction calculated by the police, to find acceleration.
How are you allowed to do that ? You're supposed to calculate 'a' from Newton's Second Law. That equation you used : a = gf was boorowed from another specific problem (one on level ground) and apllied here. You can't do that.

Since the car is on a slope F(g) would be F(cos[theta]) for the y-direction and F(sin[theta]) for the x-direction.In the x-direction are the forces F(sin[theta]) + F(sliding friction)=ma.
Almost correct. But what about the signs ?

Did you draw the free body diagram ? Which way does Fsin(theta) act ? and in which direction does friction act ?

In the y-direction are the forces -F(cos[theta])+F(normal)=ma.
Is there an acceleration along the y-direction ?

After she slams the breaks its all sliding friction that's causing the car to travel. Since she was already in motion her initial velocity cannot be zero, but after hitting the car her final velocity would be. I could use the distance slid and the acceleration found to find the time using uniform acceleration equations and the initial velocity.
I think.
Let's say you make the corrections necessary above, and calculate the acceleration (or deceleration) of the car. So, you know 'a' and you know the distance 's'. The final velocity (after sliding, before colliding) is not zero, because if it were, there would be no collision. Can you write down the equation that relates the final velocity to the intial velocity, the distance and the acceleration ?

The others I haven't much of an idea since we just started those topics on circular motion and torque the day before this problem set was given. I am still trying to grasp those.
We can not help you with problems that you show no work on. If you want help on those problems, you must go over the concepts in your text, and give it your best shot. Tell us what you think and we can help you from there.

Problem 2 should be the first one, topically: FBD; x(t), v(t), a(t); F = ma
Problem 3 is a 2-d motion (mv^2/r) problem, do it after P4.
Problem 1 deals with rotational inertia. We'll help, but do the others first.

You're using "tilted" x and y coordinates so that a has only one component,
so the other $$\Sigma F_y = m a_y = 0$$ are simple.

1. What are the four hard problems in the problem set?

The four hard problems in the problem set are the P vs. NP problem, the Riemann Hypothesis, the Birch and Swinnerton-Dyer Conjecture, and the Hodge Conjecture. These are unsolved problems in mathematics that have been open for many years and continue to challenge mathematicians.

2. Why are these problems considered difficult?

These problems are considered difficult because they require advanced mathematical knowledge and techniques to even begin to approach a solution. Additionally, they have been open for a long time and have resisted attempts at solving them, making them some of the most challenging problems in mathematics.

3. Has any progress been made towards solving these problems?

Yes, there has been some progress made towards solving these problems, but they remain unsolved. For example, the Riemann Hypothesis has been proven for certain special cases, but a general proof has not yet been found. Similarly, the P vs. NP problem has many partial solutions, but no complete proof.

4. Why are these problems important?

These problems are important because solving them would have significant implications in various fields, such as computer science, cryptography, and number theory. It would also advance our understanding of mathematics and potentially lead to new discoveries and developments.

5. Are there any rewards or prizes for solving these problems?

Yes, there are prizes and rewards offered for solving these problems. For example, the Clay Mathematics Institute offers a \$1 million prize for solving any one of the seven Millennium Prize Problems, which includes the P vs. NP problem and the Birch and Swinnerton-Dyer Conjecture. Additionally, solving these problems would bring recognition and fame to the mathematician who solves them.

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