dynamics homework

ME2030, Dynamics: the Science of how things move

Homework 3, Spring 2018, Prof. Manoj Srinivasan

Due: Friday Jan 26, 2018. Drop o↵ in class or before.

Max: 20 points. 1-2 problems will be graded at random.

Q1. Polar coordinates. An ant is walking on a table. The ant is a bit lost, so it is wandering around aimlessly. With respect to some origin (see figure), the polar coordinates of the ant’s position is known to

be: r(t) = R(1 + cos(!t)) and ✓(t) = kt + b sin(!t), where R, k,! are constants, t is time in seconds. For numerical answers, you can use R = 0.2 m, ! = 0.1 Hz, k = 0.1 Hz, b = 0.1 and t = 9 s. a) Determine velocity v in polar coordinates. b) Determine acceleration a in polar coordinates. c) Represent unit vectors er and e✓ in terms of i and j. c) Represent velocity v in Cartesian coordinates. d) Determine acceleration a in Cartesian coordinates e) Determine velocity v in tangential-normal coordinates. i.e., in terms of et and en.

Q2. Running in a hail-storm. Little hail-stones1 are falling from the sky. They fall vertically downward, with speed 10 m/s relative to the ground. You are running fast to get away from the hail. Your velocity is

horizontal with magnitude 8 m/s, relative to the ground.

a) Using Cartesian coordinates shown in figure, represent the hail-stones’ velocity vhail and vperson as vectors. b) Compute the velocity of hail relative to you. Write as a vector.

c) Compute the velocity of you relative to the hail stone. Write as a vector.

d) Compute the magnitude of these relative velocities.

Q3. Kinematics on a boat. You are on a big boat. The boat is accelerating horizontally at 1 m/s2

relative to water. The water is not moving relative to earth. You drop a ball at time t = 0. The ball accelerates down vertically relative to earth due to gravity (g = 9.8 m/s2). a) What is the acceleration of the ball relative to the boat?

b) Say, the initial velocity of the ball relative to the boat is zero — starting at rest relative to the boat.

Determine vx(t) and vy(t) of ball relative to boat. c) Say initial velocity of boat is 5 m/s horizontal relative to the water. Determine vx(t) and vy(t) of ball relative to the water.

Q4. For the pulley system shown, a) The acceleration of A is 3 m/s

2 downward. Determine the acceleration of B.

b) The velocity of A is 2 m/s upward. Determine velocity of B.

Q5. Slipping ladder. Someone has propped a ladder onto a wall improperly. The ladder starts to slip against the ground. The horizontal velocity of point B is 1 m/s. What is the vertical (downward) velocity

of point A of the ladder, as it slips against the vertical wall? The length of the ladder is L = 5 m. Assume that angle ✓ = 60 degrees. The ladder is rigid, so that its length does not change. Hint: Use Pythagoras theorem and di↵erentiate the equation with respect to time.

1Little pieces of ice

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