Forces in Two and Three Dimensions

Practice

practice problem 1

A laboratory cart (m1 = 500 g) is pulled horizontally across a level track by a lead weight (m2 = 25 g) suspended vertically off the end of a pulley as shown in the diagram below. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.)

  1. Draw a free body diagram for …
    1. the cart
    2. the weight
  2. Determine …
    1. the acceleration of the system
    2. the tension in the string

solution

Answer it.

practice problem 2

A pendulum can be used as an inexpensive accelerometer by a passenger in a car, airplane, roller coaster, or other vehicle. When the vehicle isn't accelerating, the pendulum will hang vertically. When the vehicle is accelerating, the pendulum will hang at an angle. Let m be the mass of the pendulum bob, ℓ be its length, a be the acceleration of the vehicle, and θ be the angle the pendulum deviates from the vertical.
  1. Draw a free body diagram for the pendulum bob.
  2. Derive an equation for acceleration of the vehicle in terms of the quantities given and known constants.

solution

[INSERT IMAGE]

Start with Newton's second law of motion …

∑ F = m a

but do it twice. (Let up and forward be the positive directions.)

horizontal vertical
∑ Fx  =  max ∑ Fy  =  may
T sin θ  =  ma T cos θ − mg  =  0
      T cos θ  =  mg

Divide these two equations.

T sin θ  =  ma
T cos θ mg

Simplify using algebra and trig identities.

tan θ =  a
g

a = g tan θ

Test the equation with a few representative values. A 0° angle indicates no acceleration, since tan 0° = 0; a 45° angle corresponds to a horizontal acceleration of 1 g, since tan 45° = 1; and a 90° angle is impossible, since tan 90° = ∞.

Although the idea of using a pendulum to measure horizontal accelerations is a simple one, there really was no need to make one until people regularly started to move at speeds faster than a fast horse. The first pendulum accelerometer was built by the British mechanical engineer Frederick Lanchester in 1889. A pencil was attached to the pendulum bob so that it could automatically draw an acceleration–time graph on a piece of paper.

Sample acceleration–time graphs recorded with Lanchester's pendulum accelerometer.
a-t graph
time →		 a-t graph
← time
Train (1889): 
 		 a, b, d, e, braking;
c, starting		 Car (1905): 
 		 top and bottom, starting and stopping;
middle, starting and coasting

Lanchester was interested in the smoothness of braking systems on cars and trains. In particular, he was curious about the cause of the sudden change in motion that happens right before a braking train comes to rest.

It has been remarked that a characteristic feature of brake diagrams is the suddenness of the drop at the instant of stopping. This is a very interesting and important point, inasmuch as it is the cause of the "jerk" nearly always experienced just as a train comes to rest; it was in fact in investigating this jerk in 1888 that the idea of the pendulum accelerometer occurred to the writer. At that time it was currently supposed that the jerk was the effect of the recoil of the buffer springs after stopping; whereas a very little consideration shows that it is in reality sudden change of acceleration [emphasis original] that we recognize physiologically as "jerk," that is df/dt [for some odd reason, he chose to use f for acceleration], and not change in the direction of motion. It suggests itself in fact that the term "jerk " might well be given a scientific meaning and be defined as ds3/dt3.

The suggestion stuck.

practice problem 3

Write something different.

solution

Answer it.

practice problem 4

Write something completely different.

solution

Answer it.

The Physics Hypertextbook

  • No condition is permanent.