**Prove that centroid of a rectangle lies at the intersection of its diagonals.**

- Consider a rectangle of base length b and height h. If we take a thin strip parallel to the X-axis at a distance y from the X-axis and of infinitesimally small thickness dy then its area is given as, dA = b dy

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- The resultant of four forces which are acting at a point O as shown in Fig. 1.20.1 is along Y-axis. The magnitude of forces F1,F3 and F4 are 10 kN, 20 kN and 40 kN respectively. The angles made by10 kN, 20 kN and 40 kN with X-axis are 30°, 90° and 120° respectively. Find the magnitude and direction of force F2 if resultant is 72 kN.
- A uniform ladder 5 m long weighs 180 N. It is placed against a wall making an angle of 60º with floor. The coefficient of friction between the wall and ladder is 0.25 and between the floor and the ladder is 0.35. The ladder has to support a mass 900 N at its top. Calculate the horizontal force P to be applied to the ladder at the floor level to prevent slipping.
- If crank OA rotates with an angular velocity of omega = 12 rad/sec, determine the velocity of piston B and the angular velocity of rod AB at the instant shown in the Fig. 4.12.1.
- An automobile is accelerated at the rate of 0.8 m/sec2 as it travels from station A to station B. If the speed of the automobile is 36 km/h as it passes station A, determine the time required for automobile to reach B and its speed as it passes station B. The distance between A and B is 250 m.
- A car starts from rest on a curved road of 200 m radius and accelerates at a constant tangential acceleration of 0.5 m/sec2. Determine the distance and time which the car will travel before the total acceleration attained by it becomes 0.75 m/sec2.
- Acceleration of particle is defined by a = 21 – 21s2, where a is acceleration in m/sec2 and s is in metres. The particle starts with rest at s = 0, Determine (a) velocity when s = 1.5 m, (b) the position where velocity is again zero, (c) the position where the velocity is maximum.
- A stone is dropped into a well and is heard to strike the water after 4 seconds. Find the depth of the well if the velocity of sound is 350 m/sec.
- Acceleration of a ship moving along a straight curve varies directly as the square of its speed. If the speed drops from 3 m/sec to 1.5 m/sec in one minute, find the distance moved in this period.
- Determine the mass moment of inertia of uniform density sphere of radius 5 cm about its centroidal axes
- Derive the expression for mass moment of inertia of a sphere about centroidal axis
- Determine the mass moment of inertia of a right circular solid cone of base radius R and height h about the axis of rotation
- Calculate the mass moment of inertia of the cylinder of radius 0.5 m, height 1 m and density 2400 kg/m3 about the centroidal axis
- Find the mass moment of inertia of a hollow cylinder about its axis. The mass of cylinder is 5 kg, inner radius 10 cm, outer radius 15 cm and height 20 cm
- Derive an expression for mass moment of inertia of a solid cylinder about its longitudinal axis and its centroidal axes
- Derive the expression of mass moment of inertia of circular disc about its diametral axis.
- Find the moment of inertia of a semicircle and quarter circle.
- Find out the centroid of an L-section of 120 mm × 80 mm× 20 mm
- Determine the centroid of a semi circular segment given that a = 100 mm and alpha = 45°.
- Derive the expression for the centroid of a parabola
- Find out the centroid of area of a circular sector and also find the centroid of a semicircle
- Show that centroid of a right angled triangle lies at (b/3, h/3) where b and h are the base and height of the triangle respectively.
- Prove that centroid of a rectangle lies at the intersection of its diagonals.
- Using the principle of virtual work, determine the angle theta for which equilibrium is maintained in the mechanism shown for given values of forces P1 and P2 applied. Length of the longer links is l and that of the shorter links is l/2.
- Two weights of 8 kN and 5 kN are attached at the ends of a flexible cable. The cable passes over a pulley of diameter 1 m. The weight of the pulley is 500 N and radius of gyration is 0.5 m about its axis of rotation. Find the torque which must be applied to the pulley to raise the 8 kN weight with an acceleration of 1.2 m/sec2. Neglect the friction in the pulley.
- A uniform homogeneous cylinder rolls without slip along a horizontal level surface with a translational velocity of 20 cm/sec. If its weight is 0.1 N and its radius is 10 cm, what is its total kinetic energy ?
- A constant force of 100 N is applied as shown tangentially on a cylinder at rest, whose mass is 50 kg and radius is 10 cm, for a distance of 5 m. Determine the angular velocity of the cylinder and the velocity of its centre of mass. Assume that there is no slip.
- The speed of a flywheel rotating at 200 rpm is uniformly increased to 300 rpm in 5 seconds. Determine the work done by the driving torque and the increase in kinetic energy during this time. Take mass of the flywheel as 25 kg and its radius of gyration as 20 cm.
- A body of mass 30 kg is projected up an incline of 30° with an initial velocity of 10 m/sec. The friction coefficient between the contacting surfaces is 0.2. Determine distance travelled by the body before coming to rest.
- Two bodies of masses 80 kg and 20 kg are connected by a thread along a rough horizontal surface under the action of a force 400 N applied to the first body of mass 80 kg as shown in Fig 5.10.1. The coefficient of friction between the sliding surfaces of the bodies and plane is 0.3. Determine the acceleration of two bodies and tension in the thread using D’Alembert’s principle.
- A slender bar AB slides down a circular surface and on a horizontal surface as shown in Fig. 5.8.1. At an instant, when theta = 45°, velocity of the end A is 2 m/sec. Determine the angular velocity of the bar and the velocity of point of contact on the circular surface.
- A compound wheel rolls without slipping between two parallel plates A and B as shown in Fig. 5.7.1. At the instant A moves to the right with a velocity of 1.2 m/sec and B moves to the left with a velocity of 0.6 m/sec. Calculate the velocity of centre of wheel and the angular velocity of wheel. Take r1 = 120 mm and r2 = 360 mm.
- A particle of mass 1 kg moves in a straight line under the influence of a force which increases linearly with time at the rate of 60 N per sec. At time t = 0, the initial force may be taken as 50 N. Determine the acceleration and velocity of the particle 4 sec after it started from the rest at the origin.
- A smooth sphere moving at 10 m/sec in the direction shown in Fig. 4.35.1 collides with another smooth sphere of double its mass and moving with 5 m/sec in the direction shown. If the coefficient of restitution is 2/3, determine their velocities after collision.
- If a ball overtakes a ball of twice its mass moving 1/7th of its velocity and if the coefficient of restitution between them is 3/4, show that the first ball after striking the second ball will remain at rest.
- At a given instant the 5 kg slender bar has the motion shown in Fig. 4.31.1. Determine the angular momentum about point G (vA = 2 m/sec).
- A bullet of mass 50 gm is fired into a freely suspended target to mass 5 kg. On impact, the target moves with a velocity of 7 m/sec along with the bullet in the direction of firing. Find the velocity of bullet.
- A football of mass 200 gm is at rest. A player kicks the balls which move with a velocity of 20 m/sec at an angle of 30° with respect to ground level. Find the force exerted by the player on the ball of duration of strikes is 0.02 seconds.
- A ball is dropped from the top of a tower. If it reaches the ground with a velocity of 30 m/sec, determine the height of the tower by the conservation of energy method.
- A car of 2 ton mass starts from rest and accelerates at a uniform rate to reach a speed of 60 kmph in 20 seconds. If the frictional resistance is 600 N/ton, determine the driving power of the engine when it reaches a speed of 60 kmph.
- The x and y coordinates of the position of a particle moving in curvilinear motion are defined by x = 2 + 3(t)^2 and y = 3 + (t)^3. Determine the particle’s position, velocity and acceleration at t = 3 sec.
- A wheel that is rotating at 300 rpm attains a speed of 180 rpm after 20 seconds. Determine the acceleration of the flywheel assuming it to be uniform. Also determine the time taken to come to rest from a speed of 300 rpm if the acceleration remains the same and number of revolutions made during this time.