High School Physics : Understand Torque

Study concepts, example questions & explanations for High School Physics

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Example Questions

Example Question #1 : Circular Motion

A shop sign weighing \(\displaystyle 230N\) hangs from the end of a uniform \(\displaystyle 160N\) beam as shown. Find the horizontal and vertical forces exerted by the hinge on the beam at the wall.

 

Possible Answers:

\(\displaystyle F_{Hingex} = 5.6N F_{Hingey} = 386.1N\)

\(\displaystyle F_{Hingex} = 3.9 F_{Hingey} = 2.3N\)

\(\displaystyle F_{Hingex} = 3.9N F_{Hingey} = 270.4NN\)

\(\displaystyle F_{Hingex} = 5.6N F_{Hingey} = 3.9N\)

\(\displaystyle F_{Hingex} = 4.7N F_{Hingey} = 135.2N\)

Correct answer:

\(\displaystyle F_{Hingex} = 5.6N F_{Hingey} = 386.1N\)

Explanation:

This is a static equilibrium problem.  In order for static equilibrium to be achieved, there are three things that must be true.  First, the sum of the forces in the horizontal direction must all equal \(\displaystyle 0\).  Second, the sum of the forces in the vertical direction must all be equal to zero.  Third, the torque around a fixed axis must equal \(\displaystyle 0\).

Let us begin by summing up the forces in the vertical direction.

\(\displaystyle \Sigma F_y = F_{Hingey} + F_{Tensiony} - F_{gbeam} -F_{gsign} = 0\)

Then let us sum up the forces in the horizontal direction

\(\displaystyle \Sigma F_x = F_{Hingex} - F_{Tensionx} = 0\)

Lastly let us analyze the torque, using the hinge as the axis point where \(\displaystyle L\) is the length of the beam.

\(\displaystyle \Sigma \tau = F_{Tensiony}L - F_{gbeam}\frac{L}{2} - F_{gsign}L = 0\)

Looking at these three equations, the easiest to work with would be our torque equation as it is only missing one variable.  If we are able to find the tension in the \(\displaystyle y\) direction we would then be able to use trigonometry to determine the tension in the cable overall.

\(\displaystyle F_{Tensiony}L_{wire} - F_{gbeam}\frac{L_{beam}}{2} - F_{gsign}L_{sign} = 0\)

 

\(\displaystyle F_{Tensiony}(1.35m) - 160N(\frac{1.7m}{2}) - 230N(1.7m) = 0\)

\(\displaystyle F_{Tensiony}(1.35m) - 136Nm - 391Nm = 0\)

\(\displaystyle F_{Tensiony}(1.35m) - 527Nm = 0\)

\(\displaystyle F_{Tensiony}(1.35m) = 527Nm\)

\(\displaystyle F_{Tensiony} = 3.9N\)

We can now substitute this value back into our vertical direction equation to determine the force on the hinge in the \(\displaystyle y\) direction.

\(\displaystyle F_{Hingey} + F_{Tensiony} - F_{gbeam} -F_{gsign} = 0\)

\(\displaystyle F_{Hingey} + 3.9N - 160N - 230N = 0\)

\(\displaystyle F_{Hingey} - 386.1N = 0\)

\(\displaystyle F_{Hingey} = 386.1N\)

We can go back to our tension in our \(\displaystyle y\) direction and use trigonometry to determine the force of the tension in \(\displaystyle x\) direction.

\(\displaystyle tan\theta = \frac{F_{Tensiony}}{F_{Tensionx}}\)

\(\displaystyle tan(35) = \frac{3.9N}{F_{Tensionx}}\)

Rearrange and solve for the tension in the \(\displaystyle x\) direction.

\(\displaystyle F_{Tensionx} = \frac{3.9N}{tan(35)}\)

\(\displaystyle F_{Tensionx} = 5.6N\)

We can now go back to our second equation and substitute this value in.

\(\displaystyle F_{Hingex} - F_{Tensionx} = 0\)

\(\displaystyle F_{Hingex} -5.6N = 0\)

\(\displaystyle F_{Hingex} = 5.6N\)

 

 

 

 

Example Question #1 : Understand Torque

A pulley has a bucket of weight \(\displaystyle 12N\) hanging from the cord over a well.  The pulley has a mass of \(\displaystyle 3kg\) and radius of \(\displaystyle 30cm\).  There is a frictional torque of \(\displaystyle 1.20Nm\) at the axle.  Assume the cord has negligible mass and does not stretch or slip on the pulley.  Calculate the linear acceleration of the bucket.

 

Possible Answers:

\(\displaystyle a = 2.4m/s^2\)

\(\displaystyle a = 17.8m/s^2\)

\(\displaystyle a = 5.3m/s^2\)

\(\displaystyle a = 10.6m/s^2\)

\(\displaystyle a = 15.2m/s^2\)

Correct answer:

\(\displaystyle a = 5.3m/s^2\)

Explanation:

First let us analyze the torque that is happening on the bucket.  There is a torque from the friction on the pulley and there is a torque from the bucket pulling on the pulley.

\(\displaystyle \Sigma\tau = I\alpha = F_TR - \tau_{friction}\)

Next let us analyze the forces involved.  There is a tension force pulling up on the bucket and there is the force of gravity pulling down on the bucket.

\(\displaystyle \Sigma F = m_{bucket}a = F_T - m_{bucket}g\)

We can rearrange this to find \(\displaystyle F_T\) so that we can substitute it into our torque equation.

\(\displaystyle m_{bucket}a +m_{bucket}g = F_T\)

Now substitute this into our torque equation

\(\displaystyle I\alpha = (m_{bucket}a +m_{bucket}g)R - \tau_{friction}\)

We know that there is a relationship between acceleration and angular acceleration.

\(\displaystyle a = R\alpha\)

So we can substitute this into our equation so that only linear acceleration is present.

\(\displaystyle I\frac{a}{R} = (m_{bucket}a +m_{bucket}g)R - \tau_{friction}\)

We also know that the moment of inertia of the system is equal to the moment of inertia of the pulley plus the bucket.

\(\displaystyle I = \frac{1}{2}m_{pulley}r^2 + m_{bucket}r^2\)

We can now substitute this into our equation.

\(\displaystyle (\frac{1}{2}m_{pulley}r^2 + m_{bucket}r^2)\frac{a}{R} = (m_{bucket}a +m_{bucket}g)R - \tau_{friction}\)

 

We can now starting putting in our known variables to solve for the missing acceleration.

\(\displaystyle (\frac{1}{2}(3kg)(.3m)^2 + (1.22kg)(.3m)^2)\frac{a}{0.3m} = (1.22kg a + 12N)(.3m) - 1.2Nm\)

\(\displaystyle (0.135 +0.1098)\frac{a}{0.3} = (0.366)a +3.6 - 1.2\)

\(\displaystyle (0.2448)\frac{a}{0.3} = (0.366)a + 2.4\)

\(\displaystyle (0.816)a = (0.366)a + 2.4\)

\(\displaystyle 0.45a = 2.4\)

\(\displaystyle a = 5.3m/s^2\)

 

 

 

Example Question #61 : Circular Motion

The torque applied to a wrench is \(\displaystyle 16N*m\). If the force applied to the wrench is \(\displaystyle 40N\), how long is the wrench?

 

 

Possible Answers:

\(\displaystyle 2.5m\)

\(\displaystyle 56m\)

\(\displaystyle 24m\)

\(\displaystyle 4m\)

\(\displaystyle 0.4m\)

Correct answer:

\(\displaystyle 0.4m\)

Explanation:

The formula for torque is:

 

\(\displaystyle T=F*r\)

 

We are given the total torque and the force applied. Using these values, we can solve for the length of the wrench.

 

\(\displaystyle 16N*m=(40N)r\)

 

\(\displaystyle 16N*m40N=r\)

 

\(\displaystyle 0.4m=r\)

 

 

Example Question #3 : Circular Motion

Two children are trying to balance on a \(\displaystyle 3.2m\) see-saw. One child has a mass of \(\displaystyle 40kg\) the other has a mass of \(\displaystyle 50kg\). If the see-saw is balanced perfectly in the middle and the \(\displaystyle 40kg\) child is sitting at one end of the see-saw, how far from the center should the \(\displaystyle 50kg\) child sit so that the system is perfectly balanced?

 

Possible Answers:

\(\displaystyle 0.78m\)

\(\displaystyle 0.32m\)

\(\displaystyle 1.6m\)

\(\displaystyle 627.84m\)

\(\displaystyle 1.28m\)

Correct answer:

\(\displaystyle 1.28m\)

Explanation:

If the see-saw is \(\displaystyle 3.2m\) in total, then it has \(\displaystyle 1.6m\) on either side of the fulcrum. 

 

The question is asking us to find the equilibrium point; that means we want the net torque to equal zero.

 

\(\displaystyle Tnet=T1+T2\)

 

\(\displaystyle 0=T1+T2\)

 

\(\displaystyle -T2=T1\)

 

Now, find the torque for the first child.

 

\(\displaystyle T=F*r\)

 

We are going to use the force of gravity for the force of the child.

 

\(\displaystyle T1=(m1g)r1\)

 

\(\displaystyle T1=(40kg*-9.81ms2)*1.6m\)

\(\displaystyle T1=-627.84N*m\)

 

When thinking of torque, treat the positive/negative as being clockwise vs. counter-clockwise instead of up vs. down. In this case, child one is generating counter-clockwise torque. That means that since \(\displaystyle -T2=T1\), child two will be generating clockwise torque.

\(\displaystyle T2=(m2g)r2=-T1\)

 

\(\displaystyle 627.84N*m=(50kg*9.81ms2)*r\)

 

Solve for the radius (distance) of the second child.

 

\(\displaystyle 627.84N*m=(490.5N)*r\)

 

\(\displaystyle 627.84N*m/490.5N=r\)

 

\(\displaystyle 1.28m=r\)

Example Question #4 : Circular Motion

Two equal forces are applied to a door at the doorknob.  The first force is applied \(\displaystyle 30^o\) to the plane of the door.  The second is applied perpendicular to the door.  Which force exerts a greater torque?

 

Possible Answers:

Both exert equal non-zero torques

The first applied at a angle

Both exert zero torques

The second applied perpendicular to the door

Correct answer:

The second applied perpendicular to the door

Explanation:

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

 

\(\displaystyle \tau = F_{perpendicular}R\)

 

In this case, one force is applied perpendicular and the other at an angle.  The one that is applied at an angle, only has a small component of the total force acting in the perpendicular direction.  This component will be smaller than the overall force.  Therefore the force that is already acting perpendicular to door will provide the greatest torque.

 

Example Question #5 : Circular Motion

A heavy boy and a light girl are balanced on a massless seesaw.  If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw?

 

Possible Answers:

Nothing, the seesaw will still be balanced

The side the boy is sitting on will tilt downward

The side the girl is setting on will tilt downward

It is impossible to say without knowing the masses and the distances

Correct answer:

Nothing, the seesaw will still be balanced

Explanation:

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

 

\(\displaystyle \tau = F_{perpendicular}R\)

 

In this example the boy of mass M is a distance R away and is balancing a girl of mass m at a distance r away.

 

\(\displaystyle MgR = mgr\)

 

If both of these kids move to a distance that is one half their original distance.

 

\(\displaystyle Mg\frac{R}{2} = mg\frac{r}{2}\)

 

The half cancels out of the equation and therefore the boy and girl will still be balanced.

 

 

Example Question #2 : Understand Torque

Two equal forces are applied to a door.  The first force is applied at the midpoint of the door, the second force is applied at the doorknob.  Both forces are applied perpendicular to the door.  Which force exerts the greater torque? 

Possible Answers:

The second at the doorknob

Both exert equal non-zero torques

Both exert zero torques

The first at the midpoint

Correct answer:

The second at the doorknob

Explanation:

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

 

\(\displaystyle \tau = F_{perpendicular}R\)

 

In this case, both forces are equal to one another.  Therefore the force that is applied at the point furthest from the axis of rotation (the hinge) will have the greater torque.  In this case, the furthest distance is the doorknob.

 

Example Question #1 : Circular Motion

A child spins a top with a radius of \(\displaystyle 2cm\) with a force of \(\displaystyle 0.5N\). How much torque is generated at the edge of the top? 

Possible Answers:

\(\displaystyle 0.01N*m\)

\(\displaystyle 0.75N*m\)

\(\displaystyle 1N*m\)

\(\displaystyle 0.005N*m\)

\(\displaystyle 0.25N*m\)

Correct answer:

\(\displaystyle 0.01N*m\)

Explanation:

Torque is a force times the radius of the circle, given by the formula:

 

\(\displaystyle T=F*r\)

 

In this case, we are given the radius in centimeters, so be sure to convert to meters:

 

\(\displaystyle 2cm=0.02m\)

 

Use this radius and the given force to solve for the torque.

 

\(\displaystyle T=F*r\)

 

\(\displaystyle T=0.5N*0.02m\)

 

\(\displaystyle T=0.01N*m\)

 

Example Question #911 : High School Physics

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A shop sign weighing \(\displaystyle 230N\) hangs from the end of a uniform \(\displaystyle 160N\) beam as shown.  Find the tension in the support wire at \(\displaystyle 35^o\).

Possible Answers:

\(\displaystyle 6.8N\)

\(\displaystyle 5.6N\)

\(\displaystyle 7.4N\)

\(\displaystyle 3.9N\)

\(\displaystyle 8.1N\)

Correct answer:

\(\displaystyle 6.8N\)

Explanation:

This is a static equilibrium problem.  In order for static equilibrium to be achieved, there are three things that must be true.  First, the sum of the forces in the horizontal direction must all equal 0.  Second, the sum of the forces in the vertical direction must all be equal to zero.  Third, the torque around a fixed axis must equal \(\displaystyle 0\).

Let us begin by summing up the forces in the vertical direction.

\(\displaystyle \Sigma F_y = F_{Hingey} + F_{Tensiony} - F_{gbeam} -F_{gsign} = 0\)

Then let us sum up the forces in the horizontal direction

\(\displaystyle \Sigma F_x = F_{Hingex} - F_{Tensionx} = 0\)

Lastly let us analyze the torque, using the hinge as the axis point where \(\displaystyle L\) is the length of the beam.

\(\displaystyle \Sigma \tau = F_{Tensiony}L - F_{gbeam}\frac{L}{2} - F_{gsign}L = 0\)

Looking at these three equations, the easiest to work with would be our torque equation as it is only missing one variable.  If we are able to find the tension in the y direction we would then be able to use trigonometry to determine the tension in the cable overall.

\(\displaystyle F_{Tensiony}L_{wire} - F_{gbeam}\frac{L_{beam}}{2} - F_{gsign}L_{sign} = 0\)

\(\displaystyle F_{Tensiony}(1.35m) - 160N(\frac{1.7m}{2}) - 230N(1.7m) = 0\)

\(\displaystyle F_{Tensiony}(1.35m) - 136Nm - 391Nm = 0\)

\(\displaystyle F_{Tensiony}(1.35m) - 527Nm = 0\)

\(\displaystyle F_{Tensiony}(1.35m) = 527Nm\)

 

 

 

\(\displaystyle F_{Tensiony} = 3.9N\)

We can now use trigonometry to determine the tension force in the wire.

\(\displaystyle sin\theta = \frac{F_{Tensiony}}{F_T}\)

\(\displaystyle sin(35) = \frac{3.9N}{F_T}\)

Rearrange to get the tension force by itself.

\(\displaystyle F_T = \frac{3.9N}{sin(35)}\)

 

 

 

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