Question

1. An object of mass 1.5 kg, attached to a string, swings back and forth like a pendulum in the vertical xy-plane. At one point in the motion, when the object is on the right side of the swing, its velocity is measured to be m/s.

a) In what direction is the object moving at this point in its motion? Draw a physical picture and the path of the object, with a velocity arrow at this point in the motion.

b) Since the string is in contact with the object, they exert forces on each other. Make a reasoned argument how the object exerts a force on the string, and then use Newton’s third law to draw an arrow to represent the force that the string exerts on the object at this point in the motion.

c) Identify all other forces that are acting on the object and draw arrows to represent these forces. Explain how you know the directions of these forces.

d) Use a pictorial arrow addition to determine the net force acting on the object at this point in its motion.

e) Based on a reasoned argument of the object’s speed and direction of motion a short time later, estimate the object’s velocity at a small time later in its motion. Use a pictorial diagram to draw arrows to determine the change in the object’s momentum between these two points.

f) Based on your estimates, calculate the magnitude and direction of the change in momentum.

g) Based on your diagrams in (d) and (e), decide if Newton’s 2nd law is upheld. If not, explain what you should change in your diagrams to get agreement, and why.

Answer 1

Solution

a)

In the diagram shown below, a body of mass m is illustrated, located at point A, to the right of its equilibrium position.

When moving an arc length to the right (s), there is a tangential force that causes the body to return to its equilibrium position, therefore the velocity at point A is tangent to the trajectory of the object.

Then:

(1)

Where:

l: is the pendulum length

w: is the angular speed

Now, of the movement equation of the mass, we have:

(2)

Sustitute eq(2) in (1), we have:

Reescribing of the last equation.

Performing a dimensional analysis, we have:

b)

The following diagram illustrates the forces exerted by the object on the rope and vice versa:

Referring to the previous figure, the force exerted by the object on the rope is the radial component of the weight (action) and the reaction is the force exerted by the rope on the object, ie the tension of the rope, then

is the weigth component radial

c)

Now, in the following figure, we ilustrate the forces, on the object:

The directions of the arrows are indicated as follows:
i) the object's weight is broken down into the radial and tangential component
ii) the tension of the rope goes from the body to the point of support of the rope, bone radially

Summary

radial components:

and

tangential components:

d)

Using the parallelogram rule, we can illustrate the net force acting on the object at the indicated point of motion