A particle performing circular motion along the path of radius 2 metre with linear acceleration 5 metre per second square find i
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Answer:
The block lands 3 m from the bottom of the cliff.
Explanation:
Hi there!
(atteched find a figure representing the situation of the problem).
To solve this problem let´s use the theorem of conservation of energy.
Initially, the object has elastic (EPE) and gravitational potential energy (PE):
PE = m · g · h
EPE = 1/2 · k · x²
Where:
m = mass of the block.
g = acceleration due to gravity.
h = height.
k = spring constant.
x = compression of the spring.
At the bottom of the cliff, this total energy, minus some energy that will be dissipated by friction during the 0.4 m displacement over the frictional surface, will be converted into kinetic energy (KE).
The kinetic energy is calculated as follows:
KE = 1/2 · m · v²
Where:
m = mass of the block
v = velocity of the block.
The work done by friction (Wf) is equal to the dissipated energy:
Wf = Fr · d
Where:
Fr = friction force.
d = distance.
The friction force is calculated as follows:
Fr = μ · N = μ · m · g
Where:
N = normal force.
g = acceleration due to gravity.
Then, the final kinetic energy can be calculated as follows:
EPE + PE - Wf = KE
EPE = 1/2 · k · x²
EPE = 1/2 · 29 N/m · (0.19 m)²
EPE = 0.52 J
PE = m · g · h
PE = 7 kg · 9.8 m/s² · (2.8 m + 1m)
PE = 260.7 J
Wf = μ · m · g · d
Wf = 1.65 · 7 kg · 9.8 m/s² · 0.4 m
Wf = 45.3 J
Then:
KE = 0.52 J + 260.7 J - 45.3 J
KE = 215.9 J
Then, we can calculate the magnitude of the velocity when the block reaches the ground:
KE = 1/2 · m · v²
215.9 J = 1/2 · 7 kg · v²
v² = 215.9 J · 2 / 7 kg
v = 7.9 m/s
The time it takes the block to reach the ground from the second drop, can be calculated with the following equation:
h = h0 + v0y · t + 1/2 · g · t²
Where:
h = height at time t.
h0 = initial height.
v0y = initial vertical velocity.
g = acceleration due to gravity.
t = time.
When the block reaches the ground its height is zero. Initially, the block does not have vertical velocity, then, v0y = 0. The initial height is 1 m. Considering the upward direction as positive, the acceleration of gravity is negative:
h = h0 + v0y · t + 1/2 · g · t²
0 m = 1 m + 0 · t - 1/2 · 9.8 m/s² · t²
-1 m = -4.9 m/s² · t²
t² = -1 m / -4.9 m/s²
t = 0.45 s
The vertical velocity (vy), when the block reaches the ground can now be calculated:
vy = v0y + g · t
vy = -9.8 m/s² · 0.45 s
vy = -4.4 m/s
And now, we can finally find the horizontal velocity (vx) of the block. The magnitude of the velocity when the block reaches the ground is calcualted as follows:
v² = vx² + vy²
v² - vy² = vx²
√(v² - vy²) = vx
vx = √((7.9 m/s)² - (4.4 m/s)²)
vx = 6.6 m/s
Since there is no force accelerating the block in the horizontal direction, the horizontal velocity of the block when it lands is equal to the initial horizontal velocity. Then, we can calculate the horizontal traveled distance:
x = x0 + v · t (x0 = 0 because we consider the edge of the cliff as the origin of the frame of reference).
x = 0 + 6.6 m/s · 0.45 s
x = 3 m
The block lands 3 m from the bottom of the cliff.
Complete and Clear Question:
A person puts a few apples into the freezer at -15°C to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of 20°C, and the heat transfer coefficient on the surfaces is 8 W/m2·K. Treating the apples as 9-cm-diameter spheres and taking their properties to be 840 kg/m3,
3.81 kJ/kg·K, k = 0.418 W/m·K, and
, determine the center and surface temperatures of the apples in 1 h. Also, determine the amount of heat transfer from each apple. Solve this problem using analytical one-term approximation method (not the Heisler charts).
Answer:
Temperature at the center of the apple, T(t) = 11.215°C
Temperature at the surface of the apple, T(r,t) = 2.68°C
Amount of heat transfer from each apple, Q = 21.47 kJ
Explanation:
For clarity and easiness of expression, the calculations are handwritten and attached as a file. Check the attached files for the complete calculation.
