Eric Nicholas K's answer to Sonia's Junior College 2 H2 Maths Singapore question.
done
{{ upvoteCount }} Upvotes
clear
{{ downvoteCount * -1 }} Downvotes
Good evening Sonia! More force is needed to pull up the bag for two reasons.
Firstly, the bag increases in gravitational energy.
Secondly, as the bag goes up, the angle which the strings make with the horizontal decreases as it goes up. This causes the vertical distance between the backpack and the pivot to get smaller over time.
With this being the case, the vertical component of the string, which provides a counterforce against the weight of the backpack, now provides a lower overall tensional force in the vertical direction (if the angle between the string and the horizontal is x and the tension is T, then the vertical component of the tension is T sin x which clearly decreases in value as x decreases from an acute angle towards an angle of zero).
This means that the effects of the weight of the bag increases since the overall vertical force now increases in favour of the weight of the bag. The only way to prevent the bag at a higher height from falling is by applying a greater force to it at the point of pulling.
Firstly, the bag increases in gravitational energy.
Secondly, as the bag goes up, the angle which the strings make with the horizontal decreases as it goes up. This causes the vertical distance between the backpack and the pivot to get smaller over time.
With this being the case, the vertical component of the string, which provides a counterforce against the weight of the backpack, now provides a lower overall tensional force in the vertical direction (if the angle between the string and the horizontal is x and the tension is T, then the vertical component of the tension is T sin x which clearly decreases in value as x decreases from an acute angle towards an angle of zero).
This means that the effects of the weight of the bag increases since the overall vertical force now increases in favour of the weight of the bag. The only way to prevent the bag at a higher height from falling is by applying a greater force to it at the point of pulling.
Date Posted:
4 years ago
For the sake of discussion, let’s assume the change in GPE is negligible.
In order to hold the bag up, the rope must exert a vertical force equal to the bag’s weight. for example, if the bag weighs 100 N, then each side of the rope must exert a 50 N (upwards) force so that the net force is zero and that the system will then be in equilibrium. Furthermore, the vertical force is expressed as Fsin(theta), where F is the force along the rope and theta is the angle between the rope and the horizontal. When theta decreases, larger F is needed to hold the bag up while maintaining the vertical component to be 50 N (upwards) on each side. This explains why you need larger force to pull the backpack up as it goes higher.
By the same above explanation, because we need the net force to be zero so that the backpack will maintain its position, therefore, the two ropes can never by horizontal. As we need the system to be in equilibrium, we need vertical components of F along the ropes to provide the 50 N x 2 upward force to cancel out the 100 N weight.
Last but not least, draw a free body diagram, and vary theta. You can easily prove that the force along the ropes increases as theta decreases.
In order to hold the bag up, the rope must exert a vertical force equal to the bag’s weight. for example, if the bag weighs 100 N, then each side of the rope must exert a 50 N (upwards) force so that the net force is zero and that the system will then be in equilibrium. Furthermore, the vertical force is expressed as Fsin(theta), where F is the force along the rope and theta is the angle between the rope and the horizontal. When theta decreases, larger F is needed to hold the bag up while maintaining the vertical component to be 50 N (upwards) on each side. This explains why you need larger force to pull the backpack up as it goes higher.
By the same above explanation, because we need the net force to be zero so that the backpack will maintain its position, therefore, the two ropes can never by horizontal. As we need the system to be in equilibrium, we need vertical components of F along the ropes to provide the 50 N x 2 upward force to cancel out the 100 N weight.
Last but not least, draw a free body diagram, and vary theta. You can easily prove that the force along the ropes increases as theta decreases.
You are right, maybe with numbers like 100 N it’s easier to visualise the scenario.
Interestingly the idea is very similar (though not fully identical) to a scenario when one person tries to secure a Badminton net into the poles. The person would have to pull downwards more so that the net goes up.
Interestingly the idea is very similar (though not fully identical) to a scenario when one person tries to secure a Badminton net into the poles. The person would have to pull downwards more so that the net goes up.
360 Tutoring Program