### Question

A block *B* of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of θ above the horizontal. The normal component of the contact force acting on B has magnitude 20 N.

** (i)** Show that sin θ = 0.28. [2]

** (ii)** Find the work done by the pulling force in moving the block a distance of 5 m. [2]

**Answer/Explanation**

**(i)** [20 + 25sinθ = 2.7g]

sinθ = 0.28

** (ii)** [25 × 5 × √(1 – 0.28^{2})]

Work done is 120 J

### Question

A block B lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N, making angles of α and β respectively with the x-direction, act on B as shown in the diagram, and B is moving in the x-direction with constant speed. It is given that cos α = 0.6 and cos β = 0.8.

**(i)** Find the total work done by the forces shown in the diagram when B has moved a distance of 20 m. [2]

**(ii)** Given that the coefficient of friction between the block and the plane is \(\frac{5}{8}\) , find the weight of the block. [3]

**Answer/Explanation**

** (i)** [WD = 30 × 20 × 0.6 + 40 × 20 × 0.8]

Work done is 1000 J

** (ii) **30 × 0.6 + 40 × 0.8 – 0.625W = 0

Weight is 80 N

**Question**

The diagram shows the vertical cross-section ABCD of a surface. BC is a circular arc, and AB and CD are tangents to BC at B and C respectively. A and D are at the same horizontal level, and B and C are at heights 2.7 m and 3.0 m respectively above the level of A and D. A particle P of mass 0.2 kg is given a velocity of 8 m\( s^{−1}\) at A, in the direction of AB (see diagram). The parts of the surface containing AB and BC are smooth.**(i)** Find the decrease in the speed of P as P moves along the surface from B to C.

The part of the surface containing CD exerts a constant frictional force on P, as it moves from C to

D, and P comes to rest as it reaches D.**(ii)** Find the speed of P when it is at the mid-point of CD.

**Answer/Explanation**

**(i)** 1⁄2 mv\(_B^{2}\)

= 1⁄2\( mv_A^{2}\)– mg × 2.7and 1⁄2 m\(v_c^{2}\)

= 1⁄2 mv\(_A^{2}\)

– mg × 3

\([v_{B}^{2}\)=\ (8^{2}-20\times 2.7,v_{c}^{2}\)= \(8^{2}-20\times 3]\)

loss of speed=\(10^{\frac{1}{2}-2\)=\(1.16ms^{-1}}\)

**(ii)** Work done = 1⁄2 0.2 × 2 2 + 0.2 × g × 3

1⁄2 (0.4 + 6) = \(1/20.2v_{M}^{2}+0.2g\times 1.5\)

Speed at midpoint is 1.41 m\(s^{–1}\)