MATH142 – TUTORIAL 3 ADDITIONAL PROBLEMS11–15 August 2014

1. A box travels along an industrial conveyor. It starts from rest at A and increases itsspeed such that at = 0.2t m/s2 (where t is in seconds). Determine

(a) the time taken for the box to reach point B, and (Ans. 5.69 s)

(b) the normal and tangential components of its acceleration when it arrives at point B.(Ans. an = 5.24 m/s2, at = 1.14 m/s2)

2. A car travels at a constant speed of 108 km/h around a circular track of radius 150m. Thesurface of the track is banked at an angle of 30◦ to the horizontal. Find the componentsof the acceleration in the direction

(a) normal to the road surface, and (Ans. 3 m/s2)

(b) parallel to the road surface. (Ans. 3√3 = 5.2 m/s2)

3. The driver of the car in the previous problem suddenly applies the brakes in such a waythat the car is brought to a halt after exactly one more lap, with constant deceleration

(i.e. at =dv

dtis constant).

(a) Find the magnitude of the car’s acceleration immediately after the brakes are applied.

(Ans.3

2π

√1 + 16π2 = 6.02 m/s2)

(b) How long after the brakes are applied will the car stop? (Ans. 20π = 62.83 s)

4. A golfer hits a golf ball from point A with an initial velocity of 50 m/s at an angle of 25◦

to the horizontal. Determine the radius of curvature of the trajectory described by theball (a) at point A, (b) at the highest point of the trajectory. (Ans. 281.2 m at point A,209.3 m at highest point)

5. At the bottom of a loop in the vertical plane an airplane has a horizontal velocity of 150m/s and is speeding up at a rate of 25 m/s2. The radius of curvature of the loop is 2000m. The plane is being tracked by radar. What are the recorded values of r, r, θ and θ at

this instant? (Ans. r = 120 m/s, r = 34.85 m/s2, θ = −0.09 rad/s, θ = 0.016 rad/s2)

6. In the cam-follower mechanism below, the slotted bar rotates with a constant angularvelocity ω = 10 rad/s, and the radial position of the follower A is determined by theprofile of the stationary cam. The path of the follower is described by the polar equationr = 1+0.5 cos 2θ m. Find the velocity and acceleration of the cam-follower when θ = 30◦

in polar coordinates. (Ans. v = −8.66 er + 12.5 eθ, a = −225 er − 173 eθ)

7. As the hydraulic cylinder rotates around O, the exposed length l of the position rod isconnected by the action of oil pressure in the cylinder. If the cylinder rotates at theconstant rate θ = 60 deg/s and l is decreasing at the constant rate of 150 mm/s, calculatethe magnitude of the velocity v and the acceleration a of end B when l = 125 mm. (Ans.v = 224.2 mm/s, a = 114.4 mm/s2)