Particle on slope then horizontal

7 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section \(A B C D\) of a surface. The parts \(A B\) and \(C D\) are straight and have lengths 2.5 m and 5.2 m respectively. \(A D\) is horizontal, and \(A B\) is inclined at \(60 ^ { \circ }\) to the horizontal. The points \(B\) and \(C\) are at the same height above \(A D\). The parts of the surface containing \(A B\) and \(B C\) are smooth. A particle \(P\) is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction \(A B\), and it subsequently reaches \(D\). The particle does not lose contact with the surface during this motion.

1. Find the speed of \(P\) at \(B\).
2. Show that the maximum height of the cross-section, above \(A D\), is less than 3.2 m .
3. State briefly why \(P\) 's speed at \(C\) is the same as its speed at \(B\).
4. The frictional force acting on the particle as it travels from \(C\) to \(D\) is 1.4 N . Given that the mass of \(P\) is 0.4 kg , find the speed with which \(P\) reaches \(D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_485_874_255_638} The diagram shows the vertical cross-section \(O A B\) of a slide. The straight line \(A B\) is tangential to the curve \(O A\) at \(A\). The line \(A B\) is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The point \(O\) is 10 m higher than \(B\), and \(A B\) has length 10 m (see diagram). The part of the slide containing the curve \(O A\) is smooth and the part containing \(A B\) is rough. A particle \(P\) of mass 2 kg is released from rest at \(O\) and moves down the slide.

1. Find the speed of \(P\) when it passes through \(A\). The coefficient of friction between \(P\) and the part of the slide containing \(A B\) is \(\frac { 1 } { 12 }\). Find
2. the acceleration of \(P\) when it is moving from \(A\) to \(B\),
3. the speed of \(P\) when it reaches \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_301_1267_616_440} The diagram shows the vertical cross-section \(A B C\) of a fixed surface. \(A B\) is a curve and \(B C\) is a horizontal straight line. The part of the surface containing \(A B\) is smooth and the part containing \(B C\) is rough. \(A\) is at a height of 1.8 m above \(B C\). A particle of mass 0.5 kg is released from rest at \(A\) and travels along the surface to \(C\).

1. Find the speed of the particle at \(B\).
2. Given that the particle reaches \(C\) with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against the resistance to motion as the particle moves from \(B\) to \(C\).

4 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_489_1041_258_552} \(A B C\) is a vertical cross-section of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 4 m higher than \(B\). The part of the surface containing \(B C\) is horizontal and the distance \(B C\) is 5 m (see diagram). A particle of mass 0.8 kg is released from rest at \(A\) and slides along \(A B C\). Find the speed of the particle at \(C\) in each of the following cases.

1. The horizontal part of the surface is smooth.
2. The coefficient of friction between the particle and the horizontal part of the surface is 0.3 .

4 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).

3 A roller-coaster car (including passengers) has a mass of 840 kg . The roller-coaster ride includes a section where the car climbs a straight ramp of length 8 m inclined at \(30 ^ { \circ }\) above the horizontal. The car then immediately descends another ramp of length 10 m inclined at \(20 ^ { \circ }\) below the horizontal. The resistance to motion acting on the car is 640 N throughout the motion.

1. Find the total work done against the resistance force as the car ascends the first ramp and descends the second ramp.
2. The speed of the car at the bottom of the first ramp is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car when it reaches the bottom of the second ramp.

3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.

1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
   1. Find the kinetic energy of Simon when he reaches the point \(R\).
   2. Hence find the speed of Simon when he reaches the point \(R\).
2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
   Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

4

1. A small heavy object of mass 10 kg travels the path ABCD which is shown in Fig. 4. ABCD is in a vertical plane; CD and AEF are horizontal. The sections of the path AB and CD are smooth but section BC is rough. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-5_368_1323_402_338} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} You should assume that
   • the object does not leave the path when travelling along ABCD and does not lose energy when changing direction
   • there is no air resistance.
   Initially, the object is projected from A at a speed of \(16.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope.
   1. Show that the object gets beyond B . The section of the path BC produces a constant resistance of 14 N to the motion of the object.
   2. Using an energy method, find the velocity of the object at D . At D , the object leaves the path and bounces on the smooth horizontal ground between E and F , shown in Fig. 4. The coefficient of restitution in the collision of the object with the ground is \(\frac { 1 } { 2 }\).
   3. Calculate the greatest height above the ground reached by the object after its first bounce.
2. A car of mass 1500 kg travelling along a straight, horizontal road has a steady speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its driving force has power \(P \mathrm {~W}\). When at this speed, the power is suddenly reduced by \(20 \%\). The resistance to the car's motion, \(F \mathrm {~N}\), does not change and the car begins to decelerate at \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the values of \(P\) and \(F\).

6 A sack of beans of mass 40 kg is pulled from rest at point A up a non-uniform slope onto and along a horizontal platform. Fig. 6 shows this slope AB and the platform BC , which is a vertical distance of 12 m above A. \begin{figure}[h] \end{figure}

1. Calculate the gain in the gravitational potential energy of the sack when it is moved from A to the platform. The sack has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by the time it reaches C at the far end of the platform. The total work done against friction in moving the sack from A to C is 484 J . There are no other resistances to the sack's motion.
2. Calculate the total work done in moving the sack between the points A and C . At point C , travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sack starts to slide down a straight chute inclined at \(\alpha\) to the horizontal. Point D at the bottom of the chute is at the same vertical height as A , as shown in Fig. 6. The chute is rough and the coefficient of friction between the chute and the sack is 0.6 . During this part of the motion, again the only resistance to the motion of the sack is friction.
3. Use an energy method to calculate the value of \(\alpha\) given that the sack is travelling at \(3 \mathrm {~ms} ^ { - 1 }\) when it reaches D . For safety reasons the sack needs to arrive at D with a speed of less than \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The value of \(\alpha\) can be adjusted to try to achieve this.
4. (A) Find the range of values of \(\alpha\) which achieve a safe speed at D .
   (B) Comment on whether adjusting \(\alpha\) is a practical way of achieving a safe speed at D .

\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

1. It is given that the sloping section of the slide is smooth.
   1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
   2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) Martin, who is of mass 40 kg, is using a slide. The slide is made of two straight sections \(AB\) and \(BC\). The section \(AB\) has length 15 metres and is at an angle of \(50°\) to the horizontal. The section \(BC\) has length 2 metres and is horizontal. \includegraphics{figure_6} Martin pushes himself from \(A\) down the slide with initial speed \(1\,\text{m}\,\text{s}^{-1}\) He reaches \(B\) with speed \(5\,\text{m}\,\text{s}^{-1}\) Model Martin as a particle.

1. Find the energy lost as Martin slides from \(A\) to \(B\). [4 marks]
2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide.
   1. Show that the magnitude of this resistance force is approximately 270 N [2 marks]
   2. Determine if Martin reaches the point \(C\). [3 marks]