NYJC EJC 2026 Forces Tutorial

9814 H3 Physics (2026) 1 Forces 1 A Level S Paper N00 Q2 The figure shows a smooth, hemispherical bowl of radius r, fixed to a horizontal table. A friend gives you a uniform, smooth rod of length L and weight W, and challenges you to find an angle  (the angle the rod makes with horizontal) such that the rod will rest in equilibrium on the rounded rim of the bowl. (a) Draw a labelled diagram showing the three forces acting on the rod when it is in equilibrium. Make clear the directions and lines of action of these forces. [3] (b) By resolving these forces vertically and horizontally and by taking moments about the lower end of the rod, show that the condition for equilibrium is =4 cos2 cosrL [5] [Hint: − = +cos( ) cos cos sin sinA B A B A B ] 2 A block of mass M sits on a plane that is inclined at an angle  to the horizontal as shown in Fig 1.1. Fig 2.1 (a) Assume that the frictional force between the block and plane is large enough to keep the block at rest and the coefficient of static friction is µ. (i) Determine the horizontal components of the normal and friction forces acting on the block. (ii) Determine the angle  at which these horizontal components are a maximum. L r 

9814 H3 Physics (2026) 2 (b) A horizontal force F = Mg is applied on the block, as shown in Fig 1.2. Assume that the frictional force between the block and plane is large enough to keep the block at rest and the coefficient of static friction is µ. Fig 1.2 (i) Determine the normal and friction forces acting on the block. (ii) Show that the range of  for which the block will remain at rest is given by:   −+ +− 11 tan11 . 3 A Level S Paper N02 Q7 (a) A smooth wire XYZ is bent to form a right angle at Y and is fixed in a vertical plane, as shown in Fig. 3.1. XY makes an angle of 30° to the horizontal. Fig. 3.1 Rings A and B, of mass m1 and m2 respectively, are placed on the wire and are connected by a light thread. The rings move without friction on the wire. When the system is in equilibrium, the thread makes an angle  with XY, as shown. (i) Ring A is in equilibrium under the action of three forces. List these forces, stating the direction of each. Sketch a vector triangle of the three forces. Label the sides and angle of the triangle. Explain why it is not possible to obtain the value of  from this triangle; [5] (ii) Ring B is also in equilibrium under the action of three forces. Sketch a labeled vector triangle of these forces. [2] (iii) Use your answers in (i) and (ii) to find an expression for  in terms of m1 and m2. [3] [ 1 2 1 3tan −= m m ] F

9814 H3 Physics (2026) 3 (b) A rope is passed several times round a fixed cylinder, as shown in Fig. 3.2. Fig. 3.2. Rope and cylinder are both rough. A large pulling force Pis applied at the left -hand end of the rope. The rope can be prevented from slipping by applying a much smaller force Fat the right-hand end. Fig. 3.3 shows a small arc of a rope, subtending a small angle  at the centre of a cylinder. When this section of the rope is in equilibrium under tension T, the normal force between the rope and the surface of the cylinder has magnitude N. (i) Find an expression for N, in terms of T and . Fig. 3.3 (ii) The force at the left-hand end of the arc in Fig. 3.3 is increased to +TT . This gives rise to a frictional force, tangential to the cylinder, of magnitude μN, where μ is a constant. User your answer to (i) to show that the condition for this section of the rope to stay in equilibrium is   = TT (iii) In a particular application, the rope is wound three times round the cylinder. The constant  is 0.40. The pulling force P at the left-hand end of the rope is 3.0  104 N. Calculate the force F at the right-hand end of the rope required just to prevent movement of the rope from right to left. [5] [16 N]

9814 H3 Physics (2026) 4 4 Serway 7th Ed P12.44 A uniform rod of weight Fg and length L is supported at its ends by a frictionless trough as shown. (a) Show that the centre of gravity of the rod must be vertically over point O when the rod is in equilibrium. (b) Determine the equilibrium value of angle . [60] 5 Freedman P11.66 One end of a uniform meter stick is placed against a vertical wall. The other end is held by a lightweight cord that makes an angle  with the stick. The coefficient of static friction between the end of the meter stick and the wall is 0.40. (a) What is the maximum value the angle  can have if the stick is to remain in equilibrium? [22] (b) Let the angle  be 15. A block of the same weight as the meter stick is suspended from the stick, as shown at a distance x from the wall. What is the minimum value of x for which the stick will remain in equilibrium? [30.2 cm] (c) When  = 15, how large must the coefficient of static friction be so that the block can be attached 10 cm from the left end of the stick without causing it to slip? [0.625] 6 Freedman P11.75 Two uniform, 75.0 g marbles 2.00 cm in diameter are stacked as shown in a container that is 3.00 cm wide. (a) Find the force that the container exerts on the marbles at the points of contact A, B, and C. [0.424 N, 1.47 N, 0.424 N] (b) What force does each marble exert on each other? [0.848 N]

9814 H3 Physics (2026) 5 7 Freedman P11.90 One end of a post weighing 400 N and with height h rests on a rough horizontal surface with s = 0.30. The upper end is held by a rope fastened to the surface and making an angle of 36.9 with the post . A horizontal force F is exerted on the post as shown. (a) If the force F is applied at the midpoint of the post, what is the largest value it can have without causing the post to slip? [400 N] (b) How large can the force be without causing the post to slip if its point of application is 6 10 the way from the ground to the top of the post? [750 N] (c) Show that if the point of application of the force is too high, the post cannot be made to slip, no matter how great the force. Find the critical height for the point of application. [0.71h]

9814 H3 Physics Tutorial Solutions 1 Forces 1 A Level S Paper N00 Q2 (a) The labelled diagram is shown in Fig 1.2. below. W refers to the weight of the rod, which acts through its CG. N refers to the normal reaction force by the inner surface of the bowl on the lower end of the rod. R refers to the reaction force by the rim of the bowl on the rod. For a body to be in equilibrium, the lines of action of all the forces must meet at one point M. (b) The vertical component of N is Nsin2; the horizontal component of N is Ncos2. The vertical component of R is Rcos; the horizontal component of R is Rsin. Since the body is in equilibrium, there is no net force in either the horizontal or vertical direction. Hence, cos2 sin (1) sin2 cos (2)   = += NR N R W Eliminating N from (1) and (2), sin sin2 coscos2   +=R RW Multiplying by cos2 throughout, sin sin2 cos cos2 cos2    +=R R W Since cos(A−B)=cosAcosB+sinAsinB, cos cos2 (3)=RW Taking moments about the lower end of the rod P, anticlockwise moment of R clockwise mome nt of Wcos R 2r cos cos (4)2   =  =  LW Substituting Rcos in (3) into (4), 2 cos2 cos 2 4