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Problem p o ints
A gunner res a bullet of mass m with sp eed v 0 at an angle from the horizontal plane Assume the bullet is red from ground level The gravitational acceleration is g Ignore air drag Express your answers in terms of m g v 0 and a When do es the bullet reach its highest p oint b How high is that ab ove the ground c With what sp eed will the bullet hit the ground d What is the horizontal distance that the bullet has traveled when it hits the ground
Problem p o ints
An unknown mass m hangs from a massless string and descends with an acceleration g The other end is attached to a mass m which slides on a frictionless horizontal table The 1 2 string go es over a uniform cylinder of mass m 2 and radius R see gure The cylinder rotates ab out a horizontal axis without friction and the string do es not slip on the cylinder Express your answers in parts b c and d in terms of g m 2 and R
m 2
m 1
R
a = g/
a Draw freeb o dy diagrams for the cylinder and the two masses b What is the tension in the horizontal section of the string c What is the tension in the vertical section of the string d What is the value of the unknown mass m 1
Problem p o ints
A solid uniform disk of mass M and radius R is oscillating ab out an axis through P The axis is p erp endicular to the plane of the disk Friction at P is negligibly small and can b e ignored The distance from P to the center C of the disk is b see gure The gravitational acceleration is g
P
b
R
C
a When the displacement angle is what then is the torque relative to p oint P b What is the moment of inertia for rotation ab out the axis through P c The torque causes an angular acceleration ab out the axis through P Write down the equation of motion in terms of the angle and the angular acceleration As the disk oscillates the maximum displacement angle �max is very small and the motion is a near p erfect simple harmonic oscillation d What is the p erio d of oscillation e As the disk oscillates is there any force that the axis at P exerts on the disk Explain your answer
Problem p o ints
A particle of mass m 1 and sp eed v 1 in the xdirection collides with another particle of mass m m is at rest b efore the collision o ccurs thus v After the collision the particles have velo cities v 0 and v 0 in the xy plane in the directions of and with the xaxis see 2 2 2 1 2 1 2 gure There are no external forces Express all your answers in terms of m 1 m 2 v 1 1 and 2 v^ ’ 1 m 1 y m 1 m 2 θ 1
x v 1 v 2 = (^0) m (^2) ’ v 2 a What is the total momentum b efore the collision direction and magnitude b What is the total momentum after the collision direction and magnitude c What is the total kinetic energy b efore the collision d What is the ratio of the sp eeds v 20 v 10 e What is the magnitude sp eed of v 10
Problem p o ints
Imagine a spherical nonrotating planet of mass M and radius R The planet has no atmosphere A spacecraft of mass m m � M is launched from the surface of the planet with sp eed v 0 at an angle of � to the lo cal vertical The ro cket burn is very short Thus you may assume that when the spacecraft has a sp eed v 0 it has not yet moved any appreciable distance a The sp eed v 0 is so high that the orbit is not b ound What is the minimum sp eed for which this is the case Now imagine that the orbit is b ound and that in its subsequent orbit the spacecraft reaches a maximum distance of R from the center of the planet At this distance the sp eed is V b What is the ratio of v 0 V c What is the total energy of the spacecraft immediately after launch d What is the total energy of the spacecraft when it is farthest away from the planet e Write down one equation which would allow you to solve for v 0 in terms of M G and R we are not asking you to solve this equation
Problem p o ints
Show that if the temp erature T in the atmosphere were indep endent of altitude the pressure p as a function of altitude h would b e �^ mg^ h p p 0 e k^ T where m is the average mass of an air molecule and p 0 is the pressure at sea level
Problem p oints
A b owling ball of mass m and radius R sits on the sm o oth o o r o f a subway car If the car has a horizontal acceleration a what is the acceleration a of the ball Assume that the ball rolls without slipping The gravitational acceleration is g 1 2
