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Wize University Physics Textbook (Master) > Newton's Laws of Motion: Forces and Dynamics

Tension

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Tension

When we exert a force by a rope, a chain or a string, there is a tension force. Usually this rope is massless, so we don't have to consider it as another "body" in our free-body diagram.

  • Tension force is always parallel to the rope, chain or string and is pointing away from the object ( in other words, you can only pull with a rope/string).




Wize Tip
Tension is the same at any point of a string, chain or rope. However, two ropes, chains or strings attached to an objet might have different tensions.

  • The rope/string can be used in conjunction with pulleys which are usually weightless and frictionless in the problems. These pulleys will change the direction of the forces that are applied.






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Example: A Rope Connecting Two Blocks


Two blocks are attached by a rope and are pulled upward by force F. The mass of the lower block is double the mass of the upper block. If the tension along the rope is 5N and the system is moving with an upward acceleration of 2.5m/s22.5m/s^2 , find magnitude of force F and the mass of each block.


Solution:

Since we have two objects in motion, we can write down the second law of Newton for each of them separately.

∑of forcesF⃗y=m.ay                ∑of forcesF⃗y=2m. ay\sum_{of\ forces}^{ }\vec{F}_y=m.a_{y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\sum_{of\ forces}^{ }\vec{F}_y=2m.\ a_y

⇒  +F−T−mg=m.ay              +T−2mg=2m.ay\Rightarrow\ \ +F-T-mg=m.a_y\ \ \ \ \ \ \ \ \ \ \ \ \ \ +T-2mg=2m.a_y

⇒  +F−(5)−m(9.81)=m.(2.5)       +5−2m(9.81)=2m.(2.5)\Rightarrow\ \ +F-\left(5\right)-m\left(9.81\right)=m.\left(2.5\right)\ \ \ \ \ \ \ +5-2m\left(9.81\right)=2m.\left(2.5\right)

⇒  +F−(5)−m(9.81)=m.(2.5)       +5−2m(9.81)=2m.(2.5)\Rightarrow\ \ +F-\left(5\right)-m\left(9.81\right)=m.\left(2.5\right)\ \ \ \ \ \ \ +5-2m\left(9.81\right)=2m.\left(2.5\right)


From these two equations, we have:

⇒ F=7.46N         m=0.2kg\Rightarrow\ F=7.46N\ \ \ \ \ \ \ \ \ m=0.2kg





A 20-kg box is hanging from two ropes as shown below. What is the tension in each rope?




Extra Practice
Two blocks are connected by a light string which passes over two pulleys as shown. The string and the pulleys are massless and there is no friction. A scale located between two pulleys. The heavier block has a mass of 5kg, and the lighter block has a mess of 3kg. What is the reading on the scale (Pick the closest answer)
Two blocks of mass m and M are connected via pulley with a configuration as shown. The surfaces are frictionless. If M=15 kg and m=8 kg and θ1=30 and θ2=60\theta_1=30\ {\rm and}\ \theta_2=60
Find the acceleration
What is the tension in the string?
Two strings on a spinning rod - solving
Practice: Two Strings on Spinning Rod
The 4.00-kg block in the figure is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in the diagram and the tension in the upper string is 80.0 N.
Suspended masses
Three masses are suspended from each other as shown. Assume they have equal masses (they are given numbers 1, 2, and 3 for identification purposes only). Which rope has the highest tension? Which has the lowest tension? Why?
Method of sections + example
Determine the forces F1, F2, F3 shown below. The structure is fixed at the left to a wall. All angles are either 45 or 90 degrees. Use positive numbers for members in tension and negative numbers for members in compression.
Practice: Two masses on a pulley
If M = 10kg and m = 5kg, find the net acceleration of the masses. Assume the pulley and strings are frictionless and of negligible mass.
A 5 kg5\ kgblock is being pulled along at table by a cable (horizontally) and is accelerating along the surface at 2[ms2]2\left[\frac{m}{s^2}\right]. The coefficient of kinetic friction between the block and the table is 0.100.10.
a) Sketch the system.
b) Find the tension in the cable.
Consider the truss shown below. (On the left is a sturdy wall.) All triangles are 45-45-90 triangles.
a) Label each member as either being in compression, in tension, or a zero force member.
b) Determine the value of F1, F2, and F3.
A 10 kg mass is hanging from a rope wrapped around a cylindrical pulley. The falling mass causes the pulley to rotate. The mass of the pulley is 3 kg. What is the acceleration of the falling mass, and what is the tension of the rope?
A horizontal rod of length L and mass M has a mass m at one end. It is supported by pivot P on the wall at the left end and a cable at angle θ\theta of at the other end as shown.
a) Find the tension in the cable.
b) If the cable snaps, with what angular velocity will the rod swing down and slam into the wall? (That is, what will the angular velocity be as it collides with the wall?)
Tension and centripetal force
The 4.00-kg block in the figure is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in the diagram and the tension in the upper string is 80.0 N.
Two strings on a spinning rod - setup, approach
The 4.00-kg block in the figure is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in the diagram and the tension in the upper string is 80.0 N.
Set up the problem (a free-body diagram, find unknown angles and distances, define coordinates) and then go to the next activity. Ask yourself: how do we deal with the rpm quantity in part (b)?
Practice: Pulley
You set up a pulley system as shown. There is kinetic friction acting on the top block (μk=0.175\mu_k=0.175) which has mass 4.5 kg. The vertical block has mass 10.0 kg.
Method of joints example
Consider the structure below. The triangles are equilateral. Label the members with letters A through G. A and B are the top two members, from left to right; C, D, E, F are the vertical members, left to right; G is the bottom member.
a) Without doing any calculations, label each member as being either in tension or compression.
b) Determine F1 and F5 in terms of F.
Forces: Torque
A wooden beam of mass 150 kg and length 10m is suspended from a horizontal wall support at the left end and connected by a cable to the same wall at the right end. The cable makes an angle of 45º with the horizontal. A person of mass 42 kg stands on the beam 2.5 m from the left wall. A 25 kg child also rests on the beam, at 7m from the left wall. Calculate (a) the tension in the rope and (b) the magnitude of the force exerted by the wall on the beam.
(Hint for part (b): there are two components. We want the magnitude, but you still need to consider both parts to find the correct magnitude.)
A tent post of length L is held up by two wires connected to the ground. If the tension in the right wire is 30.0 N, what is the tension in the left wire?
A camper hangs a 100kg pack to two separate trees. The two pieces of rope he uses are of different lengths, so the angles of depression (below horizontal) are 30o and 60o. Find the tension in each rope.
A 700kg elevator accelerates upward at 5m/s2, pulled by a cable of negligible mass. Find the tension force in the cable.
Practice: Pulley (no friction)
A giant ice cube of mass 400 kg is on an icy surface (there is no friction). The ice cube is connected by a weightless pulley to a mass that is hanging off the side of the surface below. You do not know the mass of the second object, but you want to measure it.
You release the ice cube from rest and let it slide. It begins to move, accelerating at a constant rate, and you measure that it travels 7.2 cm in 10 seconds. What is the mass of the object?
A camper hangs 100.0kg pack between two trees using two separate ropes of different lengths. One rope hangs at 60.0o below the top of the trees, and the other hangs at 30.0o. Find the tension in each rope.
A 5 kg5\ kgblock is being pulled along at table by a cable and is accelerating along the surface at 2m/s2m/s. The coefficient of kinetic friction between the block and the table is 0.100.10. Draw the
system then find the tension in the cable. Does the force in the cable change if the system is frictionless? Why or why not?
Two masses on a pulley on a slope
Two blocks of mass m and M are connected via pulley with a configuration as shown. The surfaces are frictionless. If M=15 kg and m=8 kg and θ1=30 and θ2=60\theta_1=30\ {\rm and}\ \theta_2=60,
a) Find the acceleration of the system.
b) What is the tension in the string?
Practice: Pulley
You set up a pulley system as shown. There is kinetic friction acting on the top block (μk=0.175\mu_k=0.175) which has mass 4.5 kg. The vertical block has mass 10.0 kg.
(three significant numbers)
Let's say Bob weighs 50kg and Joe weighs 100kg and they are playing a game of tug of war, where the tension in the rope is 50N. Ignore friction to solve for the acceleration of the two men.
Two masses on a pulley
If M = 10.0kg and m = 5.0kg, find the net acceleration of the masses. Assume the pulley and strings are frictionless and of negligible mass. Enter your answer with 2 significant digits, units are m/s2.
Two blocks of mass m1=2kg,m2=4kgm_1 = 2kg, m_2 = 4 kg are connected over a pulley as shown below. The pulley has mass M = 1kg and radius R= 2 cm. What is the acceleration of the blocks and the tension in the rope on either side of the pulley? (HINT: The tension must be different or the pulley would not rotate.)
Inclined pully system
Two blocks are connected by a pulley system. One block (10 kg) rests on a plane inclined at 35º to the horizontal, while the other block (5.0 kg) hangs off the edge of the incline. The coefficient of friction between the block and the table is 0.30. The blocks are released from rest, what are their accelerations and what is the tension?
1) Two masses are attached together by a rope as shown in the picture. If the system is released from the rest, what is the direction of the motion and acceleration for each mass? (Imagine there is no friction between masses and the inclined surface) M2 = 2M1 = 2kg
Two blocks of mass m and M are connected via pulley with a configuration as shown. The surfaces are frictionless. If M=15 kg and m=8 kg and θ1=30 and θ2=60\theta_1=30\ {\rm and}\ \theta_2=60
Find the acceleration
What is the tension in the string?
A 700kg elevator accelerates upward at 5m/s2, pulled by a cable of negligible mass. Find the tension force in the cable. What happens if the elevator accelerates downward?
A camper hangs a 100-kg pack between two trees using separate ropes of different lengths, as shown in the figure. Find the tension in each of the ropes.
A round spherical steel ball is hanging from ceiling via two ropes as shown below. Ropes are identical and maximum tension they tolerate is 150 Newton. Find maximum weight of the ball that these ropes can handle without breaking.
An object with mass M is hanging from a massless ring that is supported by two identical ropes attached to vertical walls. The ropes can tolerate maximum tension of 1200 N. What is maximum value for M that is allowed without breaking the ropes?
A) 162 kg B) 136 kg C) 112 kg D) 151 kg
Let's say Bob weighs 50kg and Joe weighs 100kg and they are playing a game of tug of war, where the tension in the rope is 50N. Ignore friction to solve for the acceleration of the two men.
Forces: Tension
A 700kg elevator accelerates upward at 5m/s2, pulled by a cable of negligible mass. Find the tension force in the cable.
If M = 10kg and m = 5kg, find the net acceleration of the masses. Assume the pulley and strings are frictionless and of negligible mass.
A camper hangs 100kg pack between two trees using two separate ropes of different lengths. Find the tension in each rope.
Two blocks are connected by a light string which passes over two pulleys as shown. The string and the pulleys are massless and there is no friction. A scale located between two pulleys. The heavier block has a mass of 5kg, and the lighter block has a mess of 3kg. What is the reading on the scale (Pick the closest answer)
Force of friction
A 400-gram package lying on a horizontal surface is attached to a horizontal string that passes over a smooth pulley. When a mass of 200 grams is attached to the other end of the string, the package is on the point of moving. Find the coefficient of friction.
A block of mass 4 kg is suspended on two strings between two walls. The angles each string makes below the horizontal is 30°. What is the tension forces in each string? Provide an answer in Newtons and 1 decimal place.
Force of tension
A monkey of mass 10kg climbs up a vertical string suspended from a hook with an acceleration of 2m/s2 . Find tension in the string.