Exam 8: Torque and Angular Momentum

A $1.500 \mathrm{~m}$ long uniform beam of mass $30.00 \mathrm{~kg}$ is supported by a wire as shown in the figure. The beam makes an angle of 10.00 degrees with the horizontal and the wire makes an angle of 30.00 degrees with the beam. A $50.00 \mathrm{~kg}$ mass, $\mathrm{m}$ , is attached to the end of the beam. What are the vertical and horizontal components of the force of the wall on the beam at the hinge?

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless incline as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 \frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}, \mathrm{M}$ is $4.00 \mathrm{~kg}$ , and the angle is 60.0 degrees, then what is the acceleration of $m_{1}$ ?

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless incline as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $4.00 \mathrm{~kg}, \mathrm{M}$ is $4.00 \mathrm{~kg}$ , and the angle is 70.0 degrees, then what is the acceleration of $m_{1}$ ?

A $5.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 0.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $3.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 4.00 \mathrm{~m}$ , $0.00 \mathrm{~m}$ ). The center of gravity of the system of masses is

A $5.00 \mathrm{~kg}$ object has a moment of inertia of $1.20 \mathrm{~kg} \mathrm{~m}^{2}$ . What torque is needed to give the object an angular acceleration of $2.0 \mathrm{rad} / \mathrm{s}^{2}$ ?

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $3.00 \mathrm{~kg}$ and $\mathrm{M}$ is $6.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to $\mathrm{m}_{2}$ ?

A $6.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 2.00 \mathrm{~m})$ and a $5.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m}, 3.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the X-axis, perpendicular to the Z-Y plane, is

A $5.00 \mathrm{~kg}$ mass is located at ( $2.00 \mathrm{~m}, 0.00 \mathrm{~m}, 3.00 \mathrm{~m})$ and a $2.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 4.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The center of gravity of the system of masses is

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the tension in the string attached to $m_{1}$ ?

A $10 \mathrm{~kg}$ object has a moment of inertia of $1.25 \mathrm{~kg} \mathrm{~m}^{2}$ . If a torque of $2.5 \mathrm{~N} \cdot \mathrm{m}$ is applied to the object, the angular acceleration is

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $3.00 \mathrm{~kg}$ and $\mathrm{M}$ is $6.00 \mathrm{~kg}$ , then what is the magnitude of the acceleration of the masses?

A torque of $20.0 \mathrm{~N} \cdot \mathrm{m}$ is applied to a bolt. The bolt rotates through an angle of 180 degrees. The work done in turning the bolt is

A $4.00 \mathrm{~kg}$ solid sphere of radius $5.00 \mathrm{~cm}$ starts from rest and rolls without slipping down a 30.0 degree incline. The acceleration of the center of mass of the solid sphere is

A $4.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $3.00 \mathrm{~kg}$ mass is located at $(-1 \mathrm{~m}, 3.00$ $\mathrm{m}, 0.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the Y-axis, perpendicular to the X-Z plane, is

A $55 \mathrm{~kg}$ girl swings on a swing, whose seat is attached to the pivot by $2.5 \mathrm{~m}$ long rigid rods (considered to be massless in this problem). As she swings, she rises to a maximum height such that the angle of the rods with respect to the vertical is 32 degrees. What is the maximum torque on the rods due to her weight, as she moves during one cycle of her swinging from the bottom of her swing path to the highest point?

A $0.50 \mathrm{~kg}$ solid disk spins at $250 \mathrm{rpm}$ . A torque of $12.5 \mathrm{~N} \cdot \mathrm{m}$ is applied for $0.150 \mathrm{~s}$ to bring it to rest. What is the disk's radius?

A 1.5m long dowel (a cylindrical rod) is pivoted about the end so that it is a pendulum of sorts - it can freely swing in a vertical plane. The dowel has a diameter of $2.0 \mathrm{~cm}$ , and its density is $3.2 \mathrm{~g} / \mathrm{cm} 3$ . When it is positioned in its swing such that its angle with the vertical is 22.5 degrees, what is the torque on the rod about the pivot due to its weight?

A $20.0 \mathrm{~kg}$ hollow cylinder $\left(\mathrm{I}=\mathrm{MR}^{2}\right)$ has a diameter of $50.0 \mathrm{~cm}$ . The cylinder is rolling down a hill with a velocity of $5.00 \mathrm{~m} / \mathrm{s}$ . The rotational kinetic energy of the rolling cylinder is

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $20.0 \mathrm{~cm}$ and a moment of inertia of $1 \frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $3.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $6.00 \mathrm{~kg}$ and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to mass $\mathrm{m}_{2}$ ?

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $20.0 \mathrm{~cm}$ and a moment of inertia of $\frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $3.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $6.00 \mathrm{~kg}$ and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the magnitude of the acceleration of the masses?