Please help with the attached problem **Problem Description:**Dean Fabien exerts a force \( P \) on the car at an angle \( \alpha = 20^\circ \). The 1760 kg car has front-wheel drive. Prof. Schulz, the driver, spins the front wheels, and the coefficient of kinetic friction \( \mu_k = 0.02 \). [Hint: Think about what direction friction should be drawn. The car is in reverse...] The combined effect of friction and a snow pile behind the rear tires exerts a horizontal resisting force \( S \). Getting the car to move requires overcoming a resisting force \( S = 420 \, \text{N} \).Questions:a. What force \( P \) must Dean Fabien exert?b. What value of \( \alpha \) will minimize the magnitude of \( P \) that Dean Fabien would have to exert to overcome the resisting force \( S = 420 \, \text{N} \)? [Hint: Think calculus!]c. What force must he exert (at this new angle)?**Visual Explanation:**1. **Image of a Car:** - A person is seen pushing a snow-covered car. The car is positioned in front of a snow pile which offers resistance.2. **Diagram of Forces:** - A side view of the car is shown with dimensions: - Distance from the rear to the center of mass: 1.62 m - Distance from the front to the center of mass: 2.55 m - Total length from rear to front: 3.40 m - Height from the ground to the point where force \( P \) is applied: 0.90 m - Force \( P \) is shown acting at an angle \( \alpha \) from the horizontal. - The resisting force \( S \) is displayed acting horizontally at the rear.This exercise involves applying concepts of physics, specifically forces and friction, to solve problems regarding the movement of the car stuck in snow with the application of external force and optimizing that force using basic calculus principles.

a. Observing the given FBD of the car, it can be deduced that the cosine-component of the applied push (P) is its horizontal component and the sine-component is its vertical component. Also, the friction and the snow pile will oppose the car’s backward motion. Hence, the resisting force (S) must act horizontally forward. The person must exert a horizontal force that overcomes the resisting force (S) on the rear wheels as well as the resisting force on the front wheels. Let m, W, and g denote the car’s mass, its weight, and the gravitational acceleration, respectively.First, determine the normal force (N) and the kinetic friction force (f) acting on the car’s spinning front wheels. For this purpose, note that the car is not moving vertically. Hence, the net vertical force (FV) acting on it is zero. Let the upward forces be positive. Use this vertical equilibrium condition and determine N and f by applying Newton’s second law as follows: 0=∑FV0=-W+N-PsinαN=mg+Psinα ∵W=mgf=μkmg+Psinα ∵f=μkN Note that this is the kinetic friction offered by the ground to the car’s front wheels. The wheels spin in such a way that the car should move backward. In the given FBD, this would be the counter-clockwise direction. Hence, the floor exerts backward friction on the front wheels.The least amount of P that would make the car move back would just overcome the car’s horizontal equilibrium condition. Let FH represent all the horizontal forces on the car. Taking the backward direction to be positive, solve Newton’s second law for the horizontal equilibrium, and determine P as follows: 0=∑F0=Pcosα+f-S=Pcosα+μkmg+Psinα-SP=S-μkmgcosα+μksinα=420 N-0.021760 kg9.8 m/s2cos20°+0.02sin20°=79.28 Nb. Use the application of differentiation method to determine the optimum angle (α) which minimizes P. First, differentiate P with respect to α and set it equal to zero to get the extremum condition for α as follows: dPdα=ddαS-μkmgcosα+μksinα0=S-μkmg--sinα+μkcosαcosα+μksinα2sinα-μkcosα=0 S-μkmg≠0α=tan-1μk=tan-10.02≈1.15°Now, differentiate this again and put the extremum condition to check whether P is a minimum as follows: d2Pdα2=ddαS-μkmgsinα-μkcosαcosα+μksinα2=S-μkmgcosα+μksinα2cosα+μksinα-sinα-μkcosα2cosα+μksinα-sinα+μkcosαcosα+μksinα4=S-μkmg1cosα+μksinα+2sinα-μkcosα2cosα+μksinα3=S-μkmg1costan-10.02+0.02sintan-10.02+2sintan-10.02-0.02costan-10.022costan-10.02+0.02sintan-10.023=0.99S-μkmg Note that S – μkmg is positive from the earlier calculation, and 0.99 is also positive. Hence, the second derivative of P is positive. This implies that P is minimum if α is tan-1(μk) or about 1.15˚.c. Evaluate P when α is tan-1(μk) by replacing this value of α in the expression for P as follows: P=S-μkmgcostan-1μk+μksintan-1μk=420 N-0.021760 kg9.8 m/s2costan-10.02+0.02sintan-10.02=75.041.0002 N=75.02 N

Science

Physics

Dean Fabien exerts a force P on the car at an angle a = 20°. The 1760 kg car has front wheel drive. Prof. Schulz, the driver, spins the front wheels, and the coefficient of kinetic friction uk = 0.02. [Hint: Think about what direction friction should be drawn. The car is in reverse...] The combined effect of friction and a snow pile behind the rear tires exerts a horizontal resting force S. Getting the car to move requires overcoming a resisting force S = 420 N. a. What force P must Dean Fabien exert? b. What value of a will minimize the magnitude of P that Dean Fabien would have to exert to overcome the resisting force S = 420 N? [Hint: Think calculus!] What force must he exert (at this new angle)? C.

Dean Fabien exerts a force P on the car at an angle a = 20°. The 1760 kg car has front wheel drive. Prof. Schulz, the driver, spins the front wheels, and the coefficient of kinetic friction uk = 0.02. [Hint: Think about what direction friction should be drawn. The car is in reverse...] The combined effect of friction and a snow pile behind the rear tires exerts a horizontal resting force S. Getting the car to move requires overcoming a resisting force S = 420 N. a. What force P must Dean Fabien exert? b. What value of a will minimize the magnitude of P that Dean Fabien would have to exert to overcome the resisting force S = 420 N? [Hint: Think calculus!] What force must he exert (at this new angle)? C.

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Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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The theory of special relativity explains the connection between space and time to objects that move in a straight line at a constant speed. The objects move at the speed of light. When an object approaches light speed, its mass is endless and cannot move faster than light. The most crucial postulate of Einstein's special relativity is that the laws of nature do not change in each frame of reference.

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Please help with the attached problem

**Problem Description:**

Dean Fabien exerts a force \( P \) on the car at an angle \( \alpha = 20^\circ \). The 1760 kg car has front-wheel drive. Prof. Schulz, the driver, spins the front wheels, and the coefficient of kinetic friction \( \mu_k = 0.02 \). [Hint: Think about what direction friction should be drawn. The car is in reverse...] The combined effect of friction and a snow pile behind the rear tires exerts a horizontal resisting force \( S \). Getting the car to move requires overcoming a resisting force \( S = 420 \, \text{N} \).

Questions:
a. What force \( P \) must Dean Fabien exert?
b. What value of \( \alpha \) will minimize the magnitude of \( P \) that Dean Fabien would have to exert to overcome the resisting force \( S = 420 \, \text{N} \)? [Hint: Think calculus!]
c. What force must he exert (at this new angle)?

**Visual Explanation:**

1. **Image of a Car:**
- A person is seen pushing a snow-covered car. The car is positioned in front of a snow pile which offers resistance.

2. **Diagram of Forces:**
- A side view of the car is shown with dimensions:
- Distance from the rear to the center of mass: 1.62 m
- Distance from the front to the center of mass: 2.55 m
- Total length from rear to front: 3.40 m
- Height from the ground to the point where force \( P \) is applied: 0.90 m
- Force \( P \) is shown acting at an angle \( \alpha \) from the horizontal.
- The resisting force \( S \) is displayed acting horizontally at the rear.

This exercise involves applying concepts of physics, specifically forces and friction, to solve problems regarding the movement of the car stuck in snow with the application of external force and optimizing that force using basic calculus principles.

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