m 1 sin b - m 2 sin a m 1 + m 2
Solution (continued)
m 1 a =-T +m 1 g sinb ,m 2 a =T -m 2 g sina .
Let us substitute the resulting expression for acceleration into the first equation of the system:
T = m 1 g sin b - m 1 sin b - m 2 sin a m 1+ m 2
m1 g =- T + m1 gsin b. | ||||
m sinb | ||||
m1 g= m gsin b- | ||||
sin a ö=
M 1 g m 1 sin b + m 2 sin b - m 1 sin b + m 2 sin a = m 1 g m 2 sin b + m 2 sin a = |
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m 1+ m 2 | m 1+ m 2 |
|
g (sinb + sina) . |
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m 1+ m 2 |
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A3. A weightless block is fixed at the top of two inclined planes making angles = 300 and = 450 with the horizon. Weights 1 and 2 of the same mass m1 = m2 = 1 kg are connected
thread and thrown over the block. Find the acceleration a with which the weights move and the tension force of the thread T. Neglect the friction of weights 1 and 2 on inclined planes, as well as the friction in the block
Solution (continued) | ||||||||||||||||||||||||
Let's do the calculations: | ||||||||||||||||||||||||
m sinb -m sin | ||||||||||||||||||||||||
» 0.24æ | ||||||||||||||||||||||||
2 hrs |
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m1 m2 | ||||||||||||||||||||||||
gН sinb +sina | ||||||||||||||||||||||||
m 1+ m 2 | ||||||||||||||||||||||||
an inextensible thread thrown over a block with a body of mass m1 (m1 >
Given: Solution
If the mass of the block can be neglected, then
acceleration and tension can be found, | ||||||
considering | progressive | |||||
movement of goods. | second law | |||||
Newton for body 1: | ||||||
F - ? | m1 a1 | M 1 g . | ||||
The same equation in projection onto the OY 1 axis: | ||||||
m1 a1 = m1 g- T1. |
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Newton's second law for body 2: | ||||||
m2 a2 | N2 + T2 + m2 g+ Ftr . | |||||
In projections on the OX 2, OY 2 axes:
A6. A body of mass m2 moves along an inclined plane with an angle of inclination, connected
an inextensible thread thrown over a block with a body of mass m1 (m1 > m2). The coefficient of friction between the mass m2 and the inclined plane μ. Find
force acting on the block axis from the side of the plane. Neglect the masses of the block and thread. Neglect friction in the block.
Solution (continued)
m2 a2 = T2 - m2 gsin a- Ftr, 0 = N2 - m2 gcos a.
The magnitude of the sliding friction force is equal to
Ftr = mN.
From the second equation of the system:
N2 = m2 gcos a.
Ftr = mN= mm2 gcos a.
Then the first equation of the system takes the form:
m2 a2 = T2 - m2 gsin a- mm2 gcos a. |
A6. A body of mass m2 moves along an inclined plane with an angle of inclination, connected
an inextensible thread thrown over a block with a body of mass m1 (m1 > m2). The coefficient of friction between the mass m2 and the inclined plane μ. Find
force acting on the block axis from the side of the plane. Neglect the masses of the block and thread. Neglect friction in the block.
Solution (continued)
The bodies are connected by an inextensible thread, so |
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Let's denote | |||||||||
If the mass of the block can be neglected, then according to Newton’s third law
T 1= T 2.
Let's denote | |||
Let us substitute the introduced notation into (1) and (2) and write the system of equations:
m1 a= m1 g- T,
A6. A body of mass m2 moves along an inclined plane with an angle of inclination, connected
an inextensible thread thrown over a block with a body of mass m1 (m1 > m2). The coefficient of friction between the mass m2 and the inclined plane μ. Find
force acting on the block axis from the side of the plane. Neglect the masses of the block and thread. Neglect friction in the block.
Solution (continued)
m1 a= m1 g- T,
m2 a= T- m2 gsin a- mm2 gcos a.
From this system of equations we find the tension force. Divide the first equation by the second:
m1 g- T | ||||
T - m2 gsin a- mm2 gcos a | ||||
(T- m2 gsin a- mm2 gcos a) = m2 (m1 g- T) , |
||||
m1 T- m1 m2 gsin a- mm1 m2 gcos a= m1 m2 g- m2 T,
T (m1 + m2 ) = m1 m2 g(sin a+ mcos a+1 ) , T = m 1 m 2 g (sin a+ mcos a+ 1 ) .
m 1+ m 2
A6. A body of mass m2 moves along an inclined plane with an angle of inclination, connected
an inextensible thread thrown over a block with a body of mass m1 (m1 > m2). The coefficient of friction between the mass m2 and the inclined plane μ. Find
force acting on the block axis from the side of the plane. Neglect the masses of the block and thread. Neglect friction in the block.
Solution (continued)
The forces applied to the block are shown in the figure.
N0 – force acting on the axis |
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block from the plane side. |
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N 0=- (T 1 | T 2 ) , |
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(T 1 | T 2) | |||||||
We find the sum of vectors using the cosine theorem from ABC. ABC is isosceles (AB = BC, T 1 = T 2 = T)
Ð BAC=Ð BCA= b 2 .
A6. A body of mass m2 moves along an inclined plane with an angle of inclination, connected
an inextensible thread thrown over a block with a body of mass m1 (m1 > m2). The coefficient of friction between the mass m2 and the inclined plane μ. Find
force acting on the block axis from the side of the plane. Neglect the masses of the block and thread. Neglect friction in the block.
Solution (continued) | ||||||||||||||
Ð ABC= p- | 2×2 =p | |||||||||||||
N 0 =T 2 +T 2 - 2T 2 cosp -b | 2T 2 | 1- cos p -b |
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2 2sin2 | 2T sinæ p | Bö, |
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b = 2 | ||||||||||||||
2T sin | ||||||||||||||
Answer: T = m 1 m 2 g (sin a + m cos a +1), m 1+ m 2
2T sin | ||||||
Problem 13056
A board with mass m 2 = 2 kg is placed on an inclined plane with an angle of inclination to the horizon α = 35°, and a block with mass m 1 = 1 kg is placed on the board. The coefficient of friction between the block and the board is f 1 = 0.1, and between the board and the plane f 2 = 0.2. Determine: 1) acceleration of the block; 2) acceleration of the board; 3) friction coefficient f 2 ", at which the board will not move.
Problem 40511
On the top inclined plane with inclination angles of 30° and 45°, they strengthened a block in the shape of a disk with a radius of 0.1 m. A thread was thrown through the block, to the ends of which weights of masses 0.4 and 0.6 kg were attached. The coefficients of friction between the bars and the plane are the same and equal to 0.2. Find the moment of inertia of the block if it rotates with angular acceleration 0.4 rad/s 2 .
Problem 18912
From a cannon that does not have a recoil device and freely slides down an inclined plane with an angle of inclination α, a shot is fired in the horizontal direction at the moment when the cannon has passed the path s. Gun mass M, projectile mass m. What must be the speed of the projectile for the gun to stop after firing?
Problem 12555
A block with a mass of 1.5 kg rests on an inclined plane with an inclination angle of 30°. It is connected to another block of mass 1 kg by a thread thrown through a block mounted on the top of an inclined plane. The block has the shape of a disk with a mass of 0.4 kg and a radius of 0.1 m. A force of 1.5 N is applied to the first block, directed upward parallel to the inclined plane. How far will the second block fall in 2 seconds from the start of the movement? How many revolutions will the block make during this time? The coefficient of friction between the block and the inclined plane is 0.1.
Problem 17211
Bodies with masses m 1 = 5 kg and m 2 = 3 kg are connected like a weightless thread, thrown over a block of mass m = 2 kg and radius r = 10 cm, lie on conjugate inclined planes with inclination angles β = 30°. The body m2 is acted upon by a vertical force F equal to 15?. Find the tension forces of the threads, the acceleration of the loads and the speed after 2 s, if the initial speed of the bodies is 0.5 m/s. Neglect friction in the block.
Problem 17551
A body of weight P is in equilibrium on a rough inclined plane with an inclination angle of 30°. Determine the sliding friction coefficient μ.
Problem 17983
A block of mass m is pulled uniformly up an inclined plane with an angle of inclination α to the horizontal. Friction coefficient k. Find the angle β of the thread with the inclined plane at which the thread tension is minimal. What is it equal to?
Let us remember: when we talk about a smooth surface, we mean that the friction between the body and this surface can be neglected.
A body of mass m located on a smooth inclined plane is acted upon by gravity m and the force normal reaction(Fig. 19.1). 
It is convenient to direct the x-axis along the inclined plane downwards, and the y-axis – perpendicular to the inclined plane upwards (Fig. 19.1). Let us denote the angle of inclination of the plane as α.
Newton's second law equation in vector form looks like 
1. Explain why the following equations are true:
2. What is the projection of the body’s acceleration onto the x-axis?
3. Why modulus is equal normal reaction forces?
4. At what angle of inclination is the acceleration of the body at smooth plane 2 times less than the acceleration of free fall?
5. At what angle of inclination of the plane is the normal reaction force 2 times less strength gravity?
By doing next task It is useful to note that the acceleration of a body located on a smooth inclined plane does not depend on the direction initial speed bodies.
6. A puck is pushed upward along a smooth inclined plane with an angle of inclination α. Initial speed of the washer v 0 .
Which one then the path will pass puck to stop?
b) After what period of time will the puck return to its starting point?
c) At what speed will the puck return to its starting point?
7. A block of mass m is on a smooth inclined plane with an angle of inclination α.
a) What is the modulus of the force holding the block on an inclined plane if the force is directed along the inclined plane? Horizontally?
b) What is the normal reaction force when the force is directed horizontally?
2. Condition of rest of a body on an inclined plane
We will now take into account the friction force between the body and the inclined plane.
If a body is at rest on an inclined plane, it is acted upon by the force of gravity m, the normal reaction force and the static friction force (Fig. 19.2). 
The static friction force is directed upward along the inclined plane: it prevents the block from sliding. Therefore, the projection of this force onto the x-axis, directed downward along the inclined plane, is negative:
F tr.pok x = –F tr.pok
8. Explain why the following equations are true:
9. A block of mass m rests on an inclined plane with inclination angle α. The coefficient of friction between the block and the plane is μ. What is the frictional force acting on the block? Is there any extra data in the condition?
10. Explain why the condition of rest of a body on an inclined plane is expressed by the inequality
Clue. Take advantage of the fact that the static friction force satisfies the inequality F tr.pok ≤ μN.
The last inequality can be used to measure the coefficient of friction: the angle of inclination of the plane is gradually increased until the body begins to slide along it (see Fig. laboratory work 4).
11. A block lying on a board began to slide along the board when its angle of inclination to the horizon was 20º. Why coefficient is equal friction between the block and the board?
12. A brick weighing 2.5 kg lies on a board 2 m long. The coefficient of friction between the brick and the board is 0.4.
a) Which one maximum height Is it possible to lift one end of the board to prevent the brick from moving?
b) What will be the friction force acting on the brick?
The static friction force acting on a body located on an inclined plane is not necessarily directed upward along the plane. It can also be directed down along the plane!
13. A block of mass m is on an inclined plane with an angle of inclination α. The coefficient of friction between the block and the plane is equal to μ, and μ< tg α. Какую силу надо приложить к бруску вдоль наклонной плоскости, чтобы сдвинуть его вдоль наклонной плоскости:
a) down? b) up?
3. Movement of a body along an inclined plane taking into account friction
Let the body now slide down the inclined plane (Fig. 19.3). In this case, it is acted upon by a sliding friction force directed opposite to the speed of the body, that is, upward along the inclined plane. 
? 15. Draw in your notebook the forces acting on the body and explain why the following equations are valid: 
16. What is the projection of the body’s acceleration onto the x-axis?
17. A block slides down an inclined plane. The coefficient of friction between the block and the plane is 0.5. How does the speed of the block change over time if the angle of inclination of the plane is equal to:
a) 20º? b) 30º? c) 45º? d) 60º?
18. The block begins to slide along the board when it is tilted at an angle of 20º to the horizontal. What determines the coefficient of friction between the block and the board? With what magnitude and direction of acceleration will the block slide down the board inclined at an angle of 30º? 15º?
Let now the initial velocity of the body be directed upward (Fig. 19.4). 
19. Draw in your notebook the forces acting on the body and explain why the following equations are valid:
20. What is the projection of the body’s acceleration onto the x-axis?
21. The block begins to slide along the board when it is tilted at an angle of 20º to the horizontal. The block was pushed up the board. With what acceleration will it move if the board is tilted at an angle: a) 30º? b) 15º? In which of these cases will the block stop at the top point?
22. A puck was pushed up an inclined plane with an initial speed v 0. The angle of inclination of the plane is α, the coefficient of friction between the washer and the plane is μ. After some time the puck returned to starting position.
a) How long did it take the puck to move up before stopping?
b) How far did the puck go before it stopped?
c) How long after this did the puck return to its initial position?
23. After a push, the block moved up an inclined plane for 2 s and then down for 3 s before returning to its initial position. The angle of inclination of the plane is 45º.
a) How many times greater is the module of acceleration of the block when moving up than when moving down?
b) What is the coefficient of friction between the block and the plane?
Additional questions and tasks
24. A block slides without initial speed from a smooth inclined plane of height h (Fig. 19.5). The angle of inclination of the plane is α. What is the speed of the block at the end of the descent? Is there extra data here? 
25. (Galileo’s problem) A straight smooth trench is drilled in a vertical disk of radius R (Fig. 19.6). What is the time it takes for the block to slide along the entire chute from rest? Angle of inclination α, in starting moment the block is at rest. 
26. A cart rolls down a smooth inclined plane with an angle of inclination α. A tripod is installed on the trolley, on which a load is suspended on a thread. Make a drawing, depict the forces acting on the load. At what angle to the vertical is the thread when the load is at rest relative to the cart?
27. A block is located on the top of an inclined plane 2 m long and 50 cm high. The coefficient of friction between the block and the plane is 0.3.
a) With what absolute acceleration will the block move if it is pushed down along the plane?
b) What speed must be imparted to the block so that it reaches the base of the plane?
28. A body weighing 2 kg is on an inclined plane. The coefficient of friction between the body and the plane is 0.4.
a) At what angle of inclination of the plane is the greatest possible meaning frictional forces?
b) What is equal to highest value frictional forces?
c) Build approximate schedule dependence of the friction force on the angle of inclination of the plane.
Clue. If tg α ≤ μ, the static friction force acts on the body, and if tg α > μ – the sliding friction force.