Worksheet: Newton's Second Law of Motion for Rotation

In this worksheet, we will practice comparing moment of inertia, torque, and angular acceleration with mass, force, and acceleration in linear motion.

Q1:

A flywheel with a moment of inertia of 1.0×10 kg⋅m2, that has an initial angular velocity of 52.36 rad/s is brought to rest by frictional torque in a time of 120 s. What is the average frictional torque on the flywheel?

Q2:

A grindstone of mass 75.0 kg consists of a solid disk with a radius of 0.280 m. A force of magnitude 180 N is applied tangentially to the disk and negligible friction resists the motion of the grindstone.

What is the magnitude of the torque exerted by the applied force?

What is the magnitude of the angular acceleration of the grindstone due to the applied force?

A frictional force of magnitude 20.0 N opposes the applied force on the grindstone at a distance of 1.50 cm from its axis. What is the magnitude of the grindstone’s angular acceleration taking this friction into account?

Q3:

A constant torque is applied to a rigid body whose moment of inertia is 4.0 kg⋅m2 around the axis of rotation. If the wheel starts from rest and attains an angular velocity of 20.0 rad/s in 10.0 s, what is the magnitude of the applied torque?

Q4:

A block of mass 3.0 kg slides down an inclined plane at an angle of 45, as shown. The block is attached to one end of a massless tether. The other end of the tether is attached to a pulley at the top of the incline. The pulley has mass of 1.0 kg, a radius of 0.50 m, and can be approximated as a disk. The coefficient of kinetic friction of the block on the plane is 0.40. What is the magnitude of the acceleration of the block?

Q5:

An automobile is suspended so that its wheels can turn freely. Each wheel acts like a disk of mass 15.0 kg with a 0.180-m radius. The walls of each tire act like an annular ring of mass 2.00 kg that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a hoop with a mass of 10.0 kg and a radius 0.330 m. The axle acts like a rod with a 14.0 kg mass that has a 2.00-cm radius. The drive shaft acts like a rod of mass 30.0 kg that has a 3.20-cm radius. The automobile’s engine can produce 200.0 N⋅m of torque. Calculate the magnitude of the angular acceleration produced by the engine if 95.0% of this torque is applied to the drive shaft, axle, and rear wheels of a car.

Q6:

A cord is wrapped around the rim of a solid cylinder of radius 0.25 m, and a constant force of 40 N is exerted on the cord, as shown. The cylinder is mounted on frictionless bearings, and its moment of inertia is 6.0 kg⋅m2.

Calculate the angular velocity of the cylinder after 5.0 m of cord have been removed.

The 40-N applied force is replaced by an object with a weight of 40 N. What is the angular velocity of the cylinder after 5.0 m of cord have unwound?

Q7:

A flywheel with a moment of inertia 𝐼=60.0 kg⋅m2 is accelerated uniformly from rest to an angular velocity of 230 rad/s in 7.00 s by a motor.

What is the angular acceleration of the flywheel?

What is the magnitude of the torque acting on the flywheel?

Q8:

A torque of 60.0 N⋅m is applied to a grinding wheel with a moment of inertia 𝐼=15 kg⋅m2 for 60.0 s. The grinding wheel is initially at rest.

What is the angular velocity of the grinding wheel after its acceleration?

What angle does the wheel move through while the torque is applied?

Q9:

A uniform cylindrical grinding wheel of mass 25.0 kg and diameter 2.00 m is turned by an electric motor. The friction in the wheel’s bearings is negligible. The wheel is accelerated from rest to 120.0 rpm in 20.0 revolutions.

What torque must be applied to the wheel?

A tool whose coefficient of kinetic friction with the wheel is 0.60 is pressed perpendicularly against the wheel with a force of 40.0 N. What torque must be supplied by the motor to keep the wheel rotating at a constant angular velocity?

Q10:

Zorch, an archenemy of Rotation Man, decides to slow Earth’s rotation to one rotation per 25.0 h by exerting an opposing force at and parallel to the equator. Rotation Man is not immediately concerned, because he knows Zorch can only exert a force of 2.00×10 N. For how much time must Zorch push with this force to accomplish his goal? Use a value of 5.98×10 kg for the mass of the Earth and a value of 6.37×10 m for the Earth’s radius.

  • A 3 . 6 2 × 1 0 year
  • B 1 . 5 8 × 1 0 year
  • C 7 . 0 2 × 1 0 year
  • D 6 . 3 6 × 1 0 year
  • E 9 . 6 5 × 1 0 year

Q11:

A grindstone can be approximated as a disk. A grindstone has a mass of 75.0 kg, a 0.220-m radius, and is turning at 60.0 rpm. A steel axe is pressed against the grindstone with a radial force of 30.0 N. Use a value of 0.400 for the kinetic coefficient of friction between steel and stone.

Calculate the angular acceleration of the grindstone.

How many rads will the grindstone turn through before coming to rest?

Q12:

A particle has a mass of 4.0 kg. At a particular instant, the particle’s position rijk=(3.0+4.02.0)m and velocity vijk=(5.01.0+3.0)/ms, and the force acting on it Fik=(12.0+25.0)N.

What is the angular momentum of the particle about the origin?

  • A ( 4 0 + 4 . 0 + 8 6 ) i j k kg⋅m2/s
  • B ( 4 0 + 4 . 0 3 6 ) i j k kg⋅m2/s
  • C ( 3 0 + 8 . 0 + 8 6 ) i j k kg⋅m2/s
  • D ( 2 0 + 4 . 0 + 3 6 ) i j k kg⋅m2/s
  • E ( 1 0 9 . 0 8 6 ) i j k kg⋅m2/s

What is the torque on the particle about the origin?

  • A ( 1 2 0 3 4 4 5 ) i j k N⋅m
  • B ( 9 2 4 4 1 2 ) i j k N⋅m
  • C ( 1 2 0 + 9 9 4 8 ) i j k N⋅m
  • D ( 1 0 0 9 9 4 8 ) i j k N⋅m
  • E ( 1 4 0 9 6 + 2 4 ) i j k N⋅m

What is the magnitude of the time rate of change of the particle’s angular momentum at this instant?

Q13:

A baseball catcher holds his fully extended 0.520 m long arm vertically upward to catch a ball. The ball moves with a horizontal speed of 45 m/s and has a mass 0.133 kg. The catcher’s arm has a mass of 4.2 kg.

What is the angular speed of the catcher’s arm immediately after catching the ball, as measured from the arm socket?

What magnitude torque is applied if the catcher stops the rotation of his arm 0.52 s after catching the ball?

Q14:

An antique car is started by exerting a force of 290 N on its crank for a time of 0.82 s, changing the force’s direction continuously to maximize the torque produced by it. The handle of the crank is 0.45 m from the pivot. What is the magnitude of the angular momentum given to the engine?

Q15:

A boat’s motor has a power output of 90.0 kW. The motor’s propellers rotate at 300 rpm. What is the magnitude of the torque on the propeller shaft?

Q16:

A flywheel that rotates about a fixed axis has a moment of inertia of 30.0 kg⋅m2. The flywheel is accelerated from rest and turns through exactly eight revolutions. What is the flywheel’s angular speed as it completes the eighth revolution?

Q17:

The fan blades of a jet engine have a moment of inertia of 30.0 kg⋅m2. The blades accelerate from rest to a rotation rate of 20 rev/s in a 10-second time interval. From this rotation rate, the blades are brought to rest in a time interval of 20 s.

What magnitude of constant torque is applied to the blades while their angular speed increases?

What magnitude of constant torque is applied to the blades while their angular speed decreases?

Q18:

A merry-go-round has a mass of 200 kg and a radius of 1.50 m. A man pushes the merry-go-round, exerting a force of 250 N at a point on its perimeter perpendicularly to its radius. In modeling the motion of the merry-go-round, consider it to be a solid disk that rotates without friction.

If there is no one on the merry-go-round, what magnitude of angular acceleration is produced by the push?

If a child of mass 18.0 kg is seated 1.25 m from the merry-go-round’s center, what magnitude of angular acceleration is produced by the push?

Q19:

A pendulum in the shape of a rod is released from rest at an angle of 30 from the vertical, as shown in the diagram. The pendulum has a length of 30 cm and a mass of 300 g. What is the pendulum’s angular speed when its moving end is at its least vertically upward displacement from the ground?

Q20:

When kicking a football, a soccer player rotates his leg about the hip joint. The velocity of the tip of the player’s shoe is 40.0 m/s and the hip joint is 1.15 m from the tip of the shoe. The shoe is in contact with the initially stationary 0.400 kg football for 15.0 ms, and the kick gives the ball a velocity of 30.0 m/s.

What is the shoe tip’s angular velocity?

What average force is exerted on the football?

Find the maximum range of the football, neglecting air resistance. Use a value of 9.8 m/s2 for the acceleration due to gravity.

Q21:

The propeller of a nuclear submarine has a moment of inertia of 800.0 kg⋅m2. The propeller is submerged and operating at full power, and its rotation rate is 4.0 rev/s. The power supply to the propeller is suddenly interrupted and the propeller does 50 kJ of work on the surrounding water without any energy being supplied to it. What is the propeller’s rotation rate after doing this work?

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