Rotational Collisions and Static Equilibrium

8.01 W12D1

Next Reading Assignment: W13D1

Young and Freedman: 10.7

2

Conditions for Static Equilibrium

(1) Translational equilibrium: the sum of the forces acting on the rigid body is zero.

(2) Rotational Equilibrium: the vector sum of the torques about any point S in a rigid body is zero.

total 1 2 ...= + + =F F F 0

,1 ,2 ...S Sτ τ τ= + + =totalS 0

Center of Mass The gravitational force between the Earth and the body is exerted at the center of mass

For a rigid body with two point particles:

1 1 2 2

1 2cm

m mm m

+=

+r rR

1cm

1

i N

i iii N

ii

m

m

=

==

=

=∑

∑

rR

For a rigid body with n- point particles:

Problem Solving Strategy Force: 1.  Identify System and draw all forces and where they act on Free

Body Force Diagram

2.  Write down equations for static equilibrium of the forces: sum of forces is zero

Torque: 1.  Choose point to analyze the torque about.

2.  Choose sign convention for torque

3.  Calculate torque about that point for each force. (Note sign of torque.)

4.  Write down equation corresponding to condition for static equilibrium: sum of torques is zero

Concept Question: Tipping A box, with its center-of-

mass off-center as indicated by the dot, is placed on an inclined plane. In which of the four orientations shown, if any, does the box tip over?

Concept Question A 1 kg rock is suspended by a massless string from one end of a 1 m measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the 0.25 m from the left end?

1. 0.25 kg. 2. 0.5 kg. 3. 1.0 kg. 4. 2.0 kg. 5. 4.0 kg. 6. Impossible to determine.

Table Problem: Forearm

You are able to support a ball of mass mb when you hold out your arm in an outstretched horizontal position thanks to the action of the tendon. The forearm and hand have a mass m. The center of mass of the forearm and hand is a distance s from the point where the upper arm meets the elbow, the tendon attaches to the forearm a distance d from the point where the upper arm meets the elbow, and the angle the tendon makes with the horizontal is α. The ball is a distance 2s from the point where the upper arm meets the elbow. The upper arm also exerts a force on the elbow. The direction and magnitude of this force depends on the other parameters (that are fixed) m, d, α , s, and mb

a)  Draw a free body diagram for all the forces that are acting on the forearm and hand. Indicate on your free body diagram your choice of unit vectors.

b) Choose a point about which to calculate the torques acting on the forearm and hand. Explain why you decided on that point.

c) What is the tension T in the tendon?

Table Problem: Person on Ladder

A person of mass mp is standing on a rung, one third of the way up a ladder of length d. The mass of the ladder is ml and is uniformly distributed. The ladder is initially inclined at an angle θ with respect to the horizontal. Assume that there is no friction between the ladder and the wall but that there is friction between the base of the ladder and the floor with a coefficient of static friction µs. In this problem you will try to find the minimum coefficient of friction µs between the ladder and the floor so that the person and ladder do not slip.

Rotational Collisions

Concept Question: A long narrow uniform stick lies motionless on

ice (assume the ice provides a frictionless surface). The center of mass of the stick is the same as the geometric center (at the midpoint of the stick). A puck (with putty on one side) slides without spinning on the ice toward the stick, hits one end of the stick, and attaches to it.

Which quantities are constant?

1.  Angular momentum of puck about center of mass of stick.

2.  Momentum of stick and ball. 3.  Angular momentum of stick and ball about any

point. 4.  Mechanical energy of stick and ball. 5.  None of the above 1-4. 6.  Three of the above 1.4 7.  Two of the above 1-4.

Worked Example A long narrow uniform stick of length l and mass m lies motionless on ice (assume the ice provides a frictionless surface). The center of mass of the stick is the same as the geometric center (at the midpoint of the stick). The moment of inertia of the stick about its center of mass is lcm. A puck (with putty on one side) has the same mass m as the stick. The puck slides without spinning on the ice with a speed of v0 toward the stick, hits one end of the stick, and attaches to it. You may assume that the radius of the puck is much less than the length of the stick so that the moment of inertia of the puck about its center of mass is negligible compared to lcm.

What is the angular velocity of the stick plus puck after the collision?

How far does the stick's center of mass move during one rotation of the stick?

Table Problem

An object of mass m and speed v0 strikes a rigid uniform rod of length lr and mass mr that is hanging by a frictionless pivot from the ceiling. Immediately after striking the rod, the object continues forward but its speed decreases to v0/2. The moment of inertia of the rod about its center of mass is I = (1/12)mr lr 2. Gravity acts with acceleration g downward. For what value of v0 will the rod just touch the ceiling on its first swing?

Table Problem Two point-like objects are located at the points A, and B, of respective masses MA=2M and MB=M, as shown in the figure below. The two objects are initially oriented along the y-axis and connected by a rod of negligible mass of length D, forming a rigid body. A force of magnitude F acting along the x direction is applied to the object at B at t = 0 for a short time interval Δt. Neglect gravity. What is the magnitude of the angular velocity of the system after the collision?

Discussion Questions Is it possible to apply another force of magnitude F along the positive x direction to prevent the system from rotating? Does it matter where the force is applied?

Is it possible to apply another force of magnitude F in some direction to prevent the center of mass from translating? Does it matter where the force is applied?