Students asked to measure the temperature of a reaction beaker recorded the following temperatures. 103.7°C, 108.4°C, 105.8°C, 1
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The total time it takes the sound to reach our on top of the cliff (10.0 s) is actually sum of two times:
- the time it takes the rock to hit the ground starting from the top of the cliff, t1
- the time it takes the sound to reach the top of the cliff starting from the ground, t2
1) The rock moves by uniformly accelerated motion, with constant acceleration
. Its law of motion is given by 
, where 
is the initial velocity of the rock, and h is the height of the cliff. The time t1 is the time at which the rock reaches the ground, so that y(t1)=0, and the equation becomes
2) The sound moves from the ground to the top of the cliff by uniform motion, with constant speed v=343 m/s. Therefore, the sound covers the distance h (the height of the cliff) in a time t2 given by
3) If we rewrite h in both equations, we can write:
(1)
4) We also know that the sum of t1 and t2 is equal to 10 seconds:
from which we find
if we substitute this into eq.(1), we get
Numerically:
Solving the equation, we find the solution
(the other solution is negative, so it does not have physical meaning). As a consequence,
and the height of the cliff is given by
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- the time it takes the rock to hit the ground starting from the top of the cliff, t1
- the time it takes the sound to reach the top of the cliff starting from the ground, t2
1) The rock moves by uniformly accelerated motion, with constant acceleration
2) The sound moves from the ground to the top of the cliff by uniform motion, with constant speed v=343 m/s. Therefore, the sound covers the distance h (the height of the cliff) in a time t2 given by
3) If we rewrite h in both equations, we can write:
4) We also know that the sum of t1 and t2 is equal to 10 seconds:
from which we find
if we substitute this into eq.(1), we get
Numerically:
Solving the equation, we find the solution
and the height of the cliff is given by
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Answer:
the equivalent mass :
the equation of the motion of the block of mass in terms of its displacement is =
Explanation:
Let use m₁ to represent the mass of the block and m₂ to represent the mass of the cylinder
The radius of the cylinder be = R
The distance between the center of the pulley to center of the block to be = x
Also, the angles of inclinations of the cylinder and the block with respect to the ground to be and
respectively.
The velocity of the block to be = v
The equivalent mass of the system =
In the terms of the equivalent mass, the kinetic energy of the system can be written as:
--------------- equation (1)
The angular velocity of the cylinder = : &
The inertia of the cylinder about its center to be = I
The angular velocity of the cylinder can be written as:
The kinetic energy of the system in terms of individual mass can be written as:
By replacing with
; we have:
------------------ equation (2)
Equating both equation (1) and (2); we have:
Therefore, the equivalent mass : which is read as;
The equivalent mass is equal to the mass of the block plus the mass of the cylinder plus the inertia by the square of the radius.
The expression for the force acting on equivalent mass due to the block is as follows:
Also; The expression for the force acting on equivalent mass due to the cylinder is as follows:
Equating the above both equations; we have the equation of motion of the equivalent system to be
which can be written as follows from the previous derivations
Finally; the equation of the motion of the block of mass in terms of its displacement is =