A plank Ab 3.0 M long weighing 20kg and with it center of gravity 2.0 M from the end A caries a load of mass 10kg at the end A it rest on two support at C and D
Compute the value of the reaction force R1 and R2
Well, you've got quite the balancing act going on here! Let's see if we can calculate those reaction forces R1 and R2.
Since the plank is in equilibrium, the sum of all the forces acting on it must be equal to zero. Let's consider the forces acting vertically.
At the center of gravity of the plank (2.0 M from end A), we have two forces acting: the weight of the plank and the reaction force R1. These forces are opposing each other. So we have:
R1 - 20kg * g = 0 (where g is the acceleration due to gravity)
Simplifying, we find:
R1 - 20kg * 9.8 m/s^2 = 0
R1 = 20kg * 9.8 m/s^2
Now, let's consider the forces acting at end A. There, we have the force of the load (10kg * g) acting downwards, and the reaction force R2 acting upwards. These forces are opposing each other as well. So we have:
R2 - 10kg * g = 0
Simplifying, we find:
R2 - 10kg * 9.8 m/s^2 = 0
R2 = 10kg * 9.8 m/s^2
Now we have our answers! R1 is equal to 196 N (approximately), and R2 is equal to 98 N (approximately).
So, in conclusion, R1 is carrying more weight (196 N) since it needs to balance the weight of the 20 kg plank, while R2 is carrying less weight (98 N) since it only needs to balance the load of 10 kg at end A.
To determine the values of the reaction forces R1 and R2, we need to calculate the torque acting on the plank and set it equal to zero.
1. Calculate the torque due to the weight of the plank:
Torque (τ1) = weight of the plank × perpendicular distance from the center of gravity to the support point
τ1 = 20 kg × 9.8 m/s^2 × 2.0 m
2. Calculate the torque due to the weight of the load:
Torque (τ2) = weight of the load × perpendicular distance from the load to the support point
τ2 = 10 kg × 9.8 m/s^2 × 3.0 m
3. Since the plank is in equilibrium, the total torque acting on the plank must be zero.
τ1 + τ2 - (R1 × 2.0 m) - (R2 × 3.0 m) = 0
4. Now we can solve for the reaction forces R1 and R2:
R1 × 2.0 m + R2 × 3.0 m = τ1 + τ2
R1 × 2.0 m + R2 × 3.0 m = (20 kg × 9.8 m/s^2 × 2.0 m) + (10 kg × 9.8 m/s^2 × 3.0 m)
2.0 R1 + 3.0 R2 = 39.2 + 29.4
5. Simplify the equation:
2.0 R1 + 3.0 R2 = 68.6
6. Since there are two unknowns (R1 and R2), we need another equation. The sum of the vertical forces must be zero in equilibrium.
R1 + R2 - (weight of the plank + weight of the load) = 0
R1 + R2 - (20 kg × 9.8 m/s^2 + 10 kg × 9.8 m/s^2) = 0
R1 + R2 - 294 N = 0
7. Simplify the equation:
R1 + R2 = 294
8. Now we have a system of linear equations:
2.0 R1 + 3.0 R2 = 68.6
R1 + R2 = 294
9. Solve the system of equations to find the values of R1 and R2. This can be done using substitution, elimination, or matrix methods.
By solving these equations, we find that R1 = 111.2 N and R2 = 182.8 N.
To compute the values of the reaction forces R1 and R2, we can use the basic principle of moments. The principle of moments states that the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about the same point.
Let's consider the points C and D as the center of moments. We will first consider point C and calculate the moments caused by the weight of the plank and the load.
1. Moments caused by the weight of the plank:
The weight of the plank acts vertically downwards at its center of gravity, which is 2.0 m from end A. The weight force can be calculated using the formula: weight = mass x gravity.
Weight of the plank = 20 kg x 9.8 m/s² (acceleration due to gravity) = 196 N
Distance of the weight from point C = 2.0 m
Moment caused by the weight of the plank = weight x distance
= 196 N x 2.0 m = 392 Nm (clockwise)
2. Moments caused by the load:
The load is located at the end A of the plank, so its distance from the point C is 3.0 m.
Moment caused by the load = load x distance
= 10 kg x 9.8 m/s² (acceleration due to gravity) x 3.0 m
= 294 Nm (clockwise)
Now, let's move on to calculating the moments about point D.
1. Moments caused by the weight of the plank:
The distance of the weight of the plank from point D is the length of the plank, 3.0 m.
Moment caused by the weight of the plank = weight x distance
= 196 N x 3.0 m = 588 Nm (anticlockwise)
2. Moments caused by the load:
The distance of the load from point D is the sum of the length of the plank (3.0 m) and the distance of the load from the end A (0 m).
Moment caused by the load = load x distance
= 10 kg x 9.8 m/s² (acceleration due to gravity) x (3.0 + 0) m
= 294 Nm (anticlockwise)
According to the principle of moments, the sum of clockwise moments is equal to the sum of anticlockwise moments.
Sum of clockwise moments = 392 Nm (from point C)
Sum of anticlockwise moments = 588 Nm + 294 Nm = 882 Nm (from point D)
To find the reaction forces R1 and R2, we can equate the magnitudes of the moments to the respective distances of the reaction forces from points C and D.
Moment caused by R1 about C = R1 x 3.0 m
Moment caused by R2 about D = R2 x 3.0 m
Setting the moments equal to each other, we can solve for R1:
392 Nm = R1 x 3.0 m
R1 = 392 Nm / 3.0 m
R1 = 130.67 N
Therefore, the value of reaction force R1 is approximately 130.67 N.
Similarly, setting the moments equal to each other, we can solve for R2:
882 Nm = R2 x 3.0 m
R2 = 882 Nm / 3.0 m
R2 = 294 N
Therefore, the value of reaction force R2 is 294 N.