Transcript: Calculating the trolley force quiz

Diagram:

A right triangle is formed by the ground, the stage, and the ramp, with the ramp as the hypotenuse sitting at 30 degree angle to the ground. Sitting on the hypotenuse of the triangle is the trolley. A vector pointing downward from the front of the trolley, parallel to the hypotenuse, represents [latex]F_p[/latex], or the force pulling the trolley down the ramp. A vector extending upward from the trolley perpendicular to the hypotenuse represents [latex]F_N[/latex], the normal force. A vector extending downward from the trolley perpendicular to the ground represents the weight force, expressed as [latex]m\times g[/latex]. The weight force and the force of the trolley on the ramp create another right triangle. The angle at which these vectors extend from the trolley is 30 degrees, and the second acute angle of the triangle is 60 degrees. The sum of the normal force and the weight force make up [latex]F_p[/latex], the force pulling the trolley down the ramp.

Question

The mass of one speaker (40kg) plus the trolley (14kg) is 54kg. Using the diagram above, where [latex]m[/latex] represents the mass, [latex]g[/latex] is gravity, and [latex]F_p[/latex] is the pulling force down the ramp, calculate the kilogram-force required to pull the speakers up a 30-degree ramp. You can assume friction is inconsequential.

Force =

Hint

In order to figure out how much force a volunteer will need to exert to pull the trolley up the ramp, we first need to know how much force is pulling the trolley down the ramp, [latex]F_p[/latex]. Our human power will need to be just greater than [latex]F_p[/latex] to pull the trolley up the ramp.

We can see from the image that [latex]F_p[/latex] is the sum of the force on the trolley from the ramp below, known as the normal force [latex](F_N)[/latex], and the weight force, represented by the mass [latex](m)[/latex] times the gravity constant [latex](g=9.8m/s^2)[/latex]. In order to calculate [latex]F_p[/latex], we can use the right triangle created by the weight force and the force of the trolley on the ramp, which is always equal and opposite to the normal force.

With right triangle trigonometry, we can determine that

[latex]\sin 30 = F_p \div mg[/latex]

[latex]F_p = mg(\sin 30)[/latex]

[latex]F_p = 54kg \times 9.8m/s^2 \times 0.5[/latex]

[latex]F_p = 264.6kgm/s^2 = 264.6N[/latex]

Remember that force is measured in newtons [latex](N)[/latex], but that’s not how we think about human effort in everyday situations. We’ll need to convert from newtons to kilogram-force, which means we need to divide our result by [latex]9.8m/s^2[/latex]:

[latex]264.6N \div 9.8m/s^2 = 27kg[/latex]

Answer

Force = 27kg