PHY_10_17_V2_DM_Conservation of Momentum

CONCEPTUAL QUESTIONS

1.      If a particle is moving with respect to a chosen origin it has linear momentum. What conditions must exist for this particle’s angular momentum to be zero about the chosen origin?

Answer

The particle must be moving on a straight line that passes through the chosen origin.

2.For a particle traveling in a straight line, are there any points about which the angular momentum is zero? Assume the line intersects the origin.

Answer

All points on the straight line will give zero angular momentum, because a vector crossed into a parallel vector is zero.

CHECK YOUR UNDERSTANDING

Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency ω₀ about the z-axis, or a solid cylinder of same mass and rotation rate about the z-axis?

Answer

Isphere=mr²,Icylinder=mr²Taking the ratio of the angular momenta, we have:

===. Thus, the cylinder has 25 more angular momentum. This is because the cylinder has more mass distributed farther from the axis of rotation.

Formative assessment - PROBLEM SOLVING

1.A bird flies overhead from where you stand at an altitude of 300.0 m and at a speed horizontal to the ground of 20.0 m/s. The bird has a mass of 2.0 kg. The radius vector to the bird makes an angle θ with respect to the ground. The radius vector to the bird and its momentum vector lie in the xy-plane. What is the bird’s angular momentum about the point where you are standing?

Answer

The magnitude of the cross product of the radius to the bird and its momentum vector yields rpsinθ, which gives rsinθ as the altitude of the bird h. The direction of the angular momentum is perpendicular to the radius and momentum vectors, which we choose arbitrarily as which is in the plane of the ground:

=×=hmv=(300.0m)(2.0kg)(20.0m/s)=12,000.0kg⋅m²/s

2.A particle of mass 5.0 kg has position vector →r=(2.0−3.0)m at a particular instant of time when its velocity is →v=(3.0)m/s with respect to the origin. (a) What is the angular momentum of the particle? (b) If a force =5.0N acts on the particle at this instant, what is the torque about the origin?

Answer

a. =45.0kg⋅m²/s

b.  =10.0N⋅m

3.An airplane of mass 4.0×104kg flies horizontally at an altitude of 10 km with a constant speed of 250 m/s relative to Earth. (a) What is the magnitude of the airplane’s angular momentum relative to a ground observer directly below the plane? (b) Does the angular momentum change as the airplane flies along its path?

Answer

a. L=1.0×10¹¹kg⋅m²/s; b. No, the angular momentum stays the same since the cross-product involves only the perpendicular distance from the plane to the ground no matter where it is along its path.

4.A boulder of mass 20 kg and radius 20 cm rolls down a hill 15 m high from rest. What is its angular momentum when it is half way down the hill? (b) At the bottom?

Answer

a. mgh=m(rω)²+mr²ω²;

ω=51.2rad/s;

L=16.4kg⋅m2/s;

b. ω=72.5rad/s

L=23.2kg⋅m²/s

5.A propeller consists of two blades each 3.0 m in length and mass 120 kg each. The propeller can be approximated by a single rod rotating about its center of mass. The propeller starts from rest and rotates up to 1200 rpm in 30 seconds at a constant rate. (a) What is the angular momentum of the propeller at t=10s;t=20s? (b) What is the torque on the propeller?

Answer

a. I=720.0kg⋅m²; α=4.20rad/s²

ω(10s)=42.0rad/s; L=3.02×104kg⋅m²/s

ω(20s)=84.0rad/s

b. τ=3.03×10³N⋅m

6.The blades of a wind turbine are 30 m in length and rotate at a maximum rotation rate of 20 rev/min. (a) If the blades are 6000 kg each and the rotor assembly has three blades, calculate the angular momentum of the turbine at this rotation rate. (b) What is the torque require to rotate the blades up to the maximum rotation rate in 5 minutes?

Answer

a. L=1.131×10⁷kg⋅m²/sL=1.131×10⁷kg·m²/s;

b. τ=3.77×10⁴N⋅m

7.A mountain biker takes a jump in a race and goes airborne. The mountain bike is travelling at 10.0 m/s before it goes airborne. If the mass of the front wheel on the bike is 750 g and has radius 35 cm, what is the angular momentum of the spinning wheel in the air the moment the bike leaves the ground?

Answer

ω=28.6rad/s⇒L=2.6kg⋅m²/s