A painter of weight 1000N stands 1.0m from the left-hand end of a uniform plank which is 5.0m long and weighs 800N. The plank is supported by two trestles each 0.5m from the opposite ends. Thre are two upward forces P and Q acting on the plank due to the trestles.
By taking moments about the left trestle, calculate the upward force exerted on the plank by the other trestle.
P = 800N + 1000N(1.0m/5.0m) = 1000N
To calculate the upward force exerted on the plank by the other trestle, we can use the principle of moments.
Let's start by calculating the total clockwise moment about the left trestle. The weight of the painter (1000 N) has a clockwise moment arm of 1.0 m, and the weight of the plank (800 N) has a clockwise moment arm of 2.5 m (half the length of the plank). Therefore, the total clockwise moment is:
Clockwise moment = (weight of painter × moment arm of painter) + (weight of plank × moment arm of plank)
= (1000 N × 1.0 m) + (800 N × 2.5 m)
= 1000 N·m + 2000 N·m
= 3000 N·m
Now, we can set up the equation for the principle of moments:
Clockwise moment = Anti-clockwise moment
3000 N·m = Upward force (P) × 0.5 m + Upward force (Q) × 4.5 m
Since we want to find the upward force exerted by the other trestle (Q), we can rearrange the equation:
Upward force (Q) × 4.5 m = 3000 N·m - Upward force (P) × 0.5 m
Finally, we divide both sides by 4.5 m to solve for Q:
Upward force (Q) = (3000 N·m - Upward force (P) × 0.5 m) / 4.5 m
Therefore, the upward force exerted on the plank by the other trestle (force Q) can be calculated using the above equation.
To solve this problem, we can use the principle of moments. The principle of moments states that for an object to be in rotational equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments about any point.
In this case, we will take moments about the left trestle. Let's denote the upward force exerted by the left trestle as P and the upward force exerted by the right trestle as Q. We need to calculate the value of Q.
First, let's calculate the clockwise and anticlockwise moments about the left trestle.
The weight of the painter acts downward and produces a clockwise moment. The moment due to the weight of the painter can be calculated as follows:
Moment due to the weight of the painter = downward force x perpendicular distance from the left trestle
= 1000N x 1.0m
= 1000 Nm (clockwise)
The weight of the plank acts downward and also produces a clockwise moment. The moment due to the weight of the plank can be calculated as follows:
Moment due to the weight of the plank = downward force x perpendicular distance from the left trestle
= 800N x 2.5m
= 2000 Nm (clockwise)
The upward force exerted by the left trestle, P, also produces a clockwise moment. The moment due to the left trestle can be calculated as follows:
Moment due to P = upward force x perpendicular distance from the left trestle
= Px 0.5m
= 0.5P Nm (clockwise)
The upward force exerted by the right trestle, Q, produces an anticlockwise moment. The moment due to the right trestle can be calculated as follows:
Moment due to Q = upward force x perpendicular distance from the left trestle
= Q x (5.0m - 0.5m)
= Q x 4.5m (anticlockwise)
Since the plank is in rotational equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments:
Sum of clockwise moments = Sum of anticlockwise moments
1000 Nm + 2000 Nm + 0.5P Nm = Q x 4.5m
Now, we can substitute the given values into the equation and solve for Q:
1000 Nm + 2000 Nm + 0.5P Nm = Q x 4.5m
3000 Nm + 0.5P Nm = Q x 4.5m
We still need the value of P to solve for Q. To calculate P, we can use the fact that the sum of the vertical forces (upward forces minus downward forces) must be equal to zero since the plank is in vertical equilibrium:
Sum of vertical forces = 0
Upward force from P + Upward force from Q - Weight of the painter - Weight of the plank = 0
P + Q - 1000N - 800N = 0
Now, we can substitute the given values into the equation and solve for P:
P + Q - 1000N - 800N = 0
P + Q = 1800N
Now that we have the value of P, we can substitute it back into the equation and solve for Q:
3000 Nm + 0.5P Nm = Q x 4.5m
3000 Nm + 0.5(1800N) Nm = Q x 4.5m
3000 Nm + 900 Nm = Q x 4.5m
3900 Nm = Q x 4.5m
Q = (3900 Nm) / (4.5m)
Q = 866.67 N
Therefore, the upward force exerted on the plank by the other trestle, Q, is approximately 866.67 N.