Welcome back to Physics 211. Today’s agenda: Review Exam 2 Potential energy of a spring Conservation of mechanical energy Conservation of total energy. Reminder …. Professor Britton Plourde Office = 223 Physics Building Office hours = Tuesday, 2-4PM, or by appointment
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Welcome back to Physics 211
Today’s agenda:
Review Exam 2
Potential energy of a spring
Conservation of mechanical energy
Conservation of total energy
Exam 2
1. [25 pts total] A block is at rest on a rough inclined plane as shown in the figure.
a. [5 pts] In the space below draw a free body diagram showing all forces on the block.
b. [5 pts] Find the components of all forces perpendicular to the plane. Hence write down an equation which ensures that the block does not accelerate in this direction.
c. [5 pts] Find the components of all forces along the plane. Hence write down an equation which ensures that the block does not accelerate down the plane.
Exam 2
1.d. [3 pts] If =300 and the mass of the block is 1 kg what is the magnitude of the friction force experienced by the block ? (take g=10 m/s2)
e. [7 pts] In the case described in part d what is the magnitude of the normal force experienced by the block. Use this value and the magnitude of the friction force computed in part d to calculate the minimum value for the coefficient of static friction necessary for the block to be at rest.
Exam 2
B
A
2. [25 pts total] A force P is applied by a hand to two blocks which are in contact on a frictionless, horizontal table as shown in the figure. The blocks accelerate to the right. Block A has a smaller mass than block B
a. [5 pts] Draw a free body diagram for block A. Label your forces with 2 subscripts indicating which body the force acts on and which body is supplying the force.
Exam 2
B
A
2.b. [5 pts] Draw a free body diagram for block B. Label your forces as in part a.
c. [4 pts] Rank the magnitudes of the normal forces on the blocks due the ground. What are their directions ? Explain your answers.
d. [4 pts] Which block experiences the larger net force ? What is the direction of this net force ? Explain your answer.
Exam 2
B
A
2.e. [2 pts] Identify which forces are Newton third law pairs
f. [5 pts] Suppose that initially the mass of block A were half that of block B. If in a subsequent experiment the mass of block A were doubled by what factor would the acceleration change assuming the pushing force remained constant ?
Exam 2
100g
3. [ 25 pts total] The figure shows a cart on a horizontal, frictionless table. The cart is connected to a massless inextensible string which passes over a frictionless pulley to a block of mass 100 g. The system is released from rest. The block is observed to be falling with an acceleration of 0.5 m/s2. Assume g=10 m/s2
a. [6 pts] Draw a free body diagram for the weight. Use it to compute the tension force on the weight.
Exam 2
100g
3.b. [6 pts] Draw a free body diagram for the cart. What is the acceleration of the cart ? Compute the mass of the cart.
c. [5 pts] What is the speed of the falling weight after 2 seconds ? How far has it fallen at that point ?
d. [4 pts] How much work has been done on the cart by the tension force after 2 seconds. What is its sign ?
e. [4 pts]Compute the kinetic energy of the cart after 2 seconds. Are the results of parts d. and e, compatible with the work-kinetic energy theorem ?
Exam 2
crate
top view
4.[25 pts total] A crate rests on the floor of a merry-go-round that rotates counterclockwise at a constant rate. The crate does not slip on the floor of the merry-go-round. Assume g=10 m/s2.
a. [4 pts] In the top view diagram at right draw arrows representing the velocity and acceleration vector for the crate.
Exam 2
crate
top view
4.b. [4 pts] Draw a free body diagram for the crate pictured from a side view. Make sure you label the forces so as to show the object on which the force acts and the object exerting the force.
c. [5 pts] The mass of the crate is 4 kg and its speed relative to the Earth is 3.0 m/s. The radius of the circle through which the crate moves is 2.0 m and the coefficient of static friction between the box and merry-go-round is =0.8. Find the magnitude of the net force on the crate. Show your work.
Exam 2
crate
top view
4.d. [5 pts] The crate is now placed closer to the center of the merry-go-round, such that the radius of the circle through which it moves is 1.0 m. The merry-go-round turns at the same rate as before. What is the new net force on the crate ?
e. [7 pts] Now the crate is placed so that the radius of its path is 4.0 m. Will it slip ? Show your working
½ mv2+mgh = constant
Piledriver
Force F=-kx (Hooke’s law)
(x extension)
W=-1/2kx2
Therefore, can define elastic (spring) potential energy
U= 1/2kx2
x
frictionless table
spring
1/2kx2+1/2mv2=constant
1/2kx02=1/2kx2+1/2mv2 or
1/2mv2= 1/2kx02-1/2kx2
vmax = (k/m)1/2x0
E=1/2mv2+U1+U2+ … is constant
Conservation of total mechanical energy
Mass hanging on spring
xeq = mg/k
z = x + xeq
(1/2)mv2 + mgx + (1/2)kx2 =
(1/2)mv2 + (1/2)kz2 - (1/2)(mg)2/k
Mass hanging on spring
oscillations between z = -x0 and z = x0
Demo
Oscillations of vertical spring and mass
Vertical spring and mass with damping
oscillates!
motion confined to
region below dotted
line
U(x)=1/2kx2
-a
a
x
Definition : DU=-FDx
or F= - DU/Dx
Force proportional to slope of U(x)
curve
Equilibrium corresponds to F=0
ie zero slope
unstable equilibrium
U(x)
E
stable equilibrium
true equilibrium
x